dissipative exciton dynamics in light-harvesting complexes

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Marco Schröter

Dissipative Exciton Dynamics in Light-Harvesting Complexes

With a foreword by Prof. Dr. Oliver Kühn

Marco SchröterInstitute of PhysicsUniversity of RostockGermany

BestMastersISBN 978-3-658-09281-8 ISBN 978-3-658-09282-5 (eBook)DOI 10.1007/978-3-658-09282-5

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Foreword

„Learning from Nature“ is a common theme in the natural sciences.The ingenious machinery of photosynthesis is particularly attractive asit has evolved to efficiently convert the sun light into chemically usableenergy. Can we mimic natural photosynthesis to open the door to anunlimited energy source? Do we need to copy just structural elementsinto man made supramolecular architectures? How important isthe interplay between structure and dynamics for the realization offunction? Finally, did Nature restrict itself to the classical laws ofPhysics or did Quantum Mechanics provide some advantage? In fact ithas been the last question which triggered most interest in the Physicscommunity recently. Initiated by sophisticated nonlinear opticalexperiments, which revealed signatures of quantum coherent evolutionat physiological temperatures, the first step of photosynthesis, i.e. theharvesting of sunlight, has been in the focus of broad research efforts.A quantum dynamical description of a system as complex as a light-

harvesting antenna protein provides quite a challenge. Models have tobe developed, flexible enough to incorporate results from experimentsas well as from atomistic simulations. Density matrix theory isthe method of choice for dynamics simulations in condensed phases.With the recent development of efficient non-Markovian and non-perturbative approaches one is in the position to follow the quantumdynamics of the electronic excitations numerically exactly, whichfacilitates a test of models without invoking further approximations.The present thesis is concerned with the dynamics and spectroscopy

of electronic energy transfer in three model systems capturing dif-ferent aspects of photosynthetic light harvesting. The simplest one,a molecular heterodimer, allows for a very detailed investigation ofcoherent oscillations in the dynamics. Here, it is possible to identify

VI Foreword

those quantum states, which are at the origin of these oscillations. Inparticular a scheme is devised, that enables discriminating betweenpure electronic and coupled electron-vibrational processes. This is ofrelevance since these types of coherences are affected differently by theprotein environment, what might help to explain the observed longlasting oscillations. The second model mimics an energy funnel, afundamental principle in photosynthesis. Special emphasis is devotedto the interplay between the energy level structure and relaxation anddecoherence dynamics. The basic features of both models come to-gether in the third application to the so-called Fenna-Matthews-Olsoncomplex, which is part of the light-harvesting apparatus of green sulfurbacteria. This heterogeneous complex hosts different energy transferpathways as well as coherent oscillations, which could be identified asbeing of electron-vibrational origin. Key to the success of this thesishas been the development of a numerical program package for thepropagation of the density matrix and the extraction of linear andnonlinear spectroscopic signals.The prospective reader of this thesis will benefit from the combi-

nation of mathematical derivations, numerical implementation, andspecific applications to current problems in excitation energy transferresearch in natural and artificial systems.

Prof. Dr. Oliver Kühn

Preface

Photosynthesis was studied intensively during the last decades bybiologists, chemists, and physicists. Although the general process iswell understood nowadays, the details, especially those concerning theeffects leading to the high efficiency of the photosynthetic apparatusof plants, bacteria, and algae, require further investigations.In the present work, an intermediate step in photosynthesis, that is

the energy transfer from the light-absorbing antenna complexes to thephotosynthetic reaction center is investigated from the perspective oftheoretical physics. The concepts of dissipation theory and excitondynamics are applied to a set of model aggregates to study variousaspects, like transfer efficiency and spectral features, of the energytransfer in light-harvesting systems.I like to thank the members of the molecular quantum dynamics

group and the dynamics of molecular systems group at the Universityof Rostock, as well as the department of chemical physics at LundUniversity, who supported me during this project. Special thanks toProf. Dr. Oliver Kühn, Prof. Dr. Tõnu Pullerits, B. Sc. Jan Schulzeand Dr. Sergei Ivanov for various discussions. Finally, I would like toacknowledge the support of my family.

Marco Schröter

Institutional Profile

„Traditio et Innovatio“ is the motto of the University of Rostock.While its history started as early as 1419, the beginnings of theInstitute of Physics date back to the late 19th century. The Institute’sgalery of forefathers includes such famous names as Stern, Schottky,Hund, and Jordan. Nowadays, the interaction of radiation withmatter is in the focus of research activities, which are linked bythe Collaborative Research Center Sfb652 „Strong Correlations andCollective Effects in Radiation Fields“. Further topics are, for instance,polymers, nanomaterials as well as surfaces and interfaces. From 2007on the University’s research profile has been shaped into four keydirections, each associated with a department under the roof of anInterdisciplinary Faculty. The Department of „Life, Light & Matter“provides the frame for many research projects of the Institute ofPhysics, which cross the traditional borders of disciplines.The research agenda of the „Molecular Quantum Dynamics“ group

headed by Prof. Oliver Kühn includes four topics. Non-reactiveand reactive dynamics of nuclear degrees of freedom, Photophysicsand Photochemistry of elementary processes, dynamics after X-raycore hole excitation, and Environmental Physics. The dynamics ofnuclei is studied in the context of linear and nonlinear vibrationalspectroscopy as a means to unravel the relation between molecularstructure, dynamics, and function. Further, laser control theory isapplied to manipulate dynamics such as to trigger, for instance, bondbreaking. The arsenal of methods comprises those from quantum,semiclassical, and classical theory. Applications are concerned, forinstance, with liquid water, ionic liquids or metal carbonyl compounds.Photoinduced processes in electronically excited molecular states areinvestigated with various electronic structure and dynamics methods.

X Institutional Profile

Particular emphasis is put on systems relevant for photocatalysis andsolar energy conversion. The present Master Thesis is an example forthe research in the context of excitation energy transfer in man-madeand natural light-harvesting antenna systems. With the advent ofnovel X-ray sources core-level spectroscopy has experienced a revivalas a means to unravel, for instance, details of electronic structureand dynamics in situ. We focus, for instance, on transition metals invarious environments, which are studied using first principles methods.Finally, an interdisciplinary effort is devoted to the introduction ofatomistic simulation techniques into the field of soil science. Targetsare pollutants and their interaction with soil components such asorganic compounds or mineral surfaces.In 2015 a new Physics building complex will be opened on the

campus of the university. It will be situated next to the researchbuilding of the Department of „Life, Light & Matter“, devoted tothe science of „Complex Molecular Systems“. We are looking forwardto the new opportunities for interdisciplinary research, which will beprovided by this excellent environment.

Contents

1. Introduction 1

2. Dissipative quantum dynamics 52.1. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Equations of motion for the reduced density matrix . . 16

2.2.1. Quantum master equation . . . . . . . . . . . . 162.2.2. Hierarchy equations of motion . . . . . . . . . 21

2.3. Spectral density models . . . . . . . . . . . . . . . . . 292.4. Calculation of optical spectra - response function for-

malism . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1. Linear Response . . . . . . . . . . . . . . . . . 402.4.2. Nonlinear Response . . . . . . . . . . . . . . . 42

3. Energy transfer in light-harvesting complexes 493.1. Dimer system . . . . . . . . . . . . . . . . . . . . . . . 503.2. Funnel-like aggregate . . . . . . . . . . . . . . . . . . . 663.3. Fenna-Matthews-Olson complex . . . . . . . . . . . . . 75

4. Summary 83

A. The Rostock HEOM package 85A.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . 85A.2. Installation and usage of the programs . . . . . . . . . 90A.3. Input documentation . . . . . . . . . . . . . . . . . . . 92

A.3.1. Run section . . . . . . . . . . . . . . . . . . . . 92A.3.2. Task section . . . . . . . . . . . . . . . . . . . . 95A.3.3. Parameter section . . . . . . . . . . . . . . . . 96A.3.4. Hamiltonian section . . . . . . . . . . . . . . . 97

XII Contents

A.3.5. Dipole section . . . . . . . . . . . . . . . . . . . 97A.3.6. Interaction section . . . . . . . . . . . . . . . . 98A.3.7. Bath section . . . . . . . . . . . . . . . . . . . 99A.3.8. Initial-RDM section . . . . . . . . . . . . . . . 99A.3.9. 2D section . . . . . . . . . . . . . . . . . . . . . 100

B. Example inputs 101B.1. Population dynamics dimer system . . . . . . . . . . . 102B.2. Linear absorption spectrum dimer system . . . . . . . 104B.3. 2D-spectrum dimer system . . . . . . . . . . . . . . . 106

C. Electronic vs. vibronic basis 109

Bibliography 113

1. Introduction

Light conversion via photosynthesis by special bacteria, algae andgreen plants is the main source of the chemical energy used by higherorganisms. The photosynthetic machinery of these organisms consistsin general of light-harvesting antenna complexes and reaction centers,which are arranged such that they form an energy funnel directing theexcitation energy flux towards the reaction center, Fig. 1.1. Sun energyabsorbed by characteristic pigments in the antenna complexes, wherean exciton is formed, is transferred to the reaction center. There, theexcitation energy is converted, via charge separation processes likeelectron transfer, to a chemical redox potential, which drives chemicalreactions. The overall quantum efficiency of the light-harvestingprocess is very high (> 95%) due to a very efficient energy transfer inthe photosynthetic complexes [1].The mechanism of excitation energy transfer in photosynthetic

complexes has been studied extensively for decades using Förster orFrenkel exciton theory [2–15]. It has caught renewed interest dueto the observation of long lasting coherent oscillations in variouslight-harvesting complexes via electronic coherent 2D-spectroscopyexperiments [16]. In particular the question, whether the observedoscillations in the 2D-spectra of light-harvesting complexes are ofelectronic or vibrational origin and their connection to the excellentenergy transfer properties, remains controversial [17–30].The protein environment and intra-molecular vibrations, which

cause, e.g., static and dynamic disorder of the excitation energiesand/or the inter-molecular couplings, create energy funnels withinthe aggregates. This leads to a spatially directed energy transferconnected with an energy relaxation towards the reaction center.Recent investigations concerning small aggregates, i.e. dimers, unveiled

M. Schröter, Dissipative Exciton Dynamics in Light-Harvesting Complexes,BestMasters, DOI 10.1007/978-3-658-09282-5_1,© Springer Fachmedien Wiesbaden 2015

2 1. Introduction

RC RC

E

RC

Figure 1.1.: Schematic picture of the photosynthetic energy transferprocess. The outer antenna complexes absorb the photon energy (redwavy arrows), which is transferred via an exciton transfer mechanism(pathways indicated by blue straight arrows) to the reaction center(RC).

that in particular vibronic couplings and resonances between electronicand vibronic levels play a distinct role for the energy transfer [22, 24,26, 27].The influence of the environment can be included into theoretical

models in the spirit of a system-bath ansatz, e.g., via a perturba-tive treatment like in the Redfield approach [31]. Such approaches,although applicable to many systems, are limited to weak interac-tions and might not cover all important effects of the system-bathinteraction. Exact methods like path integral approaches, however,are limited to rather small systems [32, 33]. The hierarchy equa-tions of motion (HEOM) formalism, initially proposed by Kubo andTanimura [34] in 1989, provides an in principle exact treatment ofthe system-bath interaction and is by virtue of recent improvementsefficient enough to investigate energy transfer processes in systemslike light-harvesting complexes [35–37].In the following section the theoretical concepts, which are used to

study the exciton dynamics in molecular aggregates embedded in anenvironment, will be introduced. First, the system-bath Hamiltonian

3

ansatz will be summarized. Second, the quantum master equation(QME) and HEOM formalisms, which provide propagation schemesfor the reduced density matrix describing the system of interest, willbe discussed. Both formalisms treat the system-bath interaction bymeans of the bath correlation function, which is usually calculatedvia the so-called spectral density, that will be introduced in Sec. 2.3.A brief summary of the response function formalism in Sec. 2.4. Itprovides the connection between the evolution of the reduced densitymatrix and the linear and nonlinear spectra of the investigated systemsis givenThe outlined theoretical concepts will be used to investigate the

dissipative exciton dynamics in dependence on the system and bathproperties of different model aggregates. First, a molecular dimersystem will be characterized in terms of population dynamics, linearabsorption, and 2D-spectra. In particular, the oscillatory features ofthe population dynamics, which might be connected to oscillations in2D-spectra, will be analysed. Second, an aggregate, which representsan energy funnel as a generic model for light-harvesting antennae,will be studied. Here, the focus is on the influence of the bathproperties on the excitation energy pathways within the aggregate.Finally, a particular example for a light-harvesting complex, theFenna-Matthews-Olson (FMO) complex, will be treated, utilizingexperimental data for the spectral density.All aforementioned examples have been studied with the HEOM

approach. The numerical implementation of this method is providedby the Rostock HEOM package, which was developed during thepresent master project. Detailed information about the package isgiven in Appendix A. The code is available upon request.

1. Introduction

2. Dissipative quantum dynamics

Complex biological systems, like the photosynthetic units of bacteriaor plants, usually consist of several pigment molecules, which formaggregates, embedded in a protein environment. Within a quantummechanical treatment it is practically impossible to take all degrees offreedom (DOFs) of such large systems explicitly into account. Notethat there exist highly efficient methods, e.g., the multiconfigurationtime-dependent Hartree method (MCTDH) [38], to treat the quantumdynamics of isolated systems, e.g., the intra-molecular dynamicsof the aforementioned pigment molecules. The recently developedmultilayer expansion of MCTDH [39, 40] facilitates the treatmentof systems with a few thousand DOFs, which is sufficient to handlesmall systems within an environment. This approach requires adiscretization of the spectral density, cf. Sec. 2.3, and is thereforerestricted in its applicability. An application of the MCTDH approachto the dissipative quantum dynamics of a light-harvesting system isgiven in Ref. [41]. However, it is preferable to reduce the dimensionalityof the problem considering only few relevant DOFs, which provideinsights into the physical processes, explicitly and to treat all otherDOFs as a heat bath interacting with the system DOFs via an energyexchange. According to this separation the Hamiltonian of the fullsystem is given by

H = Hs +Hb +Hs-b, (2.1)

where Hs/b denotes the system/bath Hamiltonian and Hs-b the inter-action between the system and the bath. The separation allows oneto solve the quantum mechanical equations of motion for the relevantsystem DOFs considering the influence of the bath DOFs exactly orapproximately.


6 2. Dissipative quantum dynamics

In the following section the Hamiltonian describing aggregatesof pigment molecules embedded in a protein environment will bediscussed. Further, two different sets of equations of motion for thereduced density matrix, namely the QME and the in principle exactHEOM will be introduced in Sec. 2.2. Both incorporate the influence ofthe bath via a spectral density function. The latter and its connectionto the Hamiltonian, as well as some models for the spectral density,will be discussed in Sec. 2.3. Finally, the optical response functionformalism, which connects the time evolution of the density matrix tothe optical spectra of the system, will be briefly introduced in Sec. 2.4.

2.1. HamiltonianThe system Hamiltonian, Hs, of an aggregate consisting of Naggmonomers is given by the Frenkel exciton Hamiltonian. It is based onthe assumption that the interacting monomers, which form the aggre-gate, retain their chemical identity. According to this assumption themonomeric adiabatic states |ma〉 can be used to construct diabatic ag-gregate states, which are coupled due to the inter-monomeric Coulombcoupling (CC). Here, a = g, e, . . . labels the adiabatic monomeric elec-tronic state andm = 1, . . . , Nagg denotes the monomer. The aggregatestates can be classified according to the number of electronic excita-tions within the aggregate. Considering only one excited state, a = e,per monomer, the aggregate ground state |g〉 and the one-excitonstates |em〉 are given by

|g〉 =∏m

|mg〉 (2.2)

and

|em〉 = |me〉∏

n�=m

|ng〉 . (2.3)

Higher-order excitations, such as the excitation of higher monomericadiabatic states, e.g., the Sn states of pigment molecules, or multiple

2.1. Hamiltonian 7

excitations within the aggregate, can be incorporated analogously intothe description. Including only the ground and one-exciton states, theFrenkel Hamiltonian is given by

Hagg =H(g)|g 〉〈 g|+∑m

Hm(em)|em 〉〈 em|

+∑m,n

Jmn(em, en)|em 〉〈 en|+Hint

=H(0)s +H(1)

s +Hint (2.4)

with the zero- and one-excition parts

H(0)s = H(g)|g 〉〈 g| (2.5)

and

H(1)s =

∑m

Hm(em)|em 〉〈 em|+∑m,n

Jmn(em, en)|em 〉〈 en|, (2.6)

respectively. The coupling elements Jmn(em, en) represent the inter-action between different one-exciton states and Hint describes, e.g.,the interaction of the system with external fields.The diagonal terms of the aggregate Hamiltonian are, according to

the definition of the exciton states, Eq. (2.2) and Eq. (2.3), given by

H(g) =∑m

H(mg) (2.7)

and

Hm(em) = H(me) +∑

n�=m

H(ng). (2.8)

Here, H(mg) and H(me) denote the Hamiltonian of the adiabaticmonomeric ground and excited state of monomer m, respectively. Thediabatic aggregate states depend on the set {R} = (R1, . . . , RNagg)of all intra-molecular nuclear coordinates, i.e. |g〉 = |g({R})〉 and|em〉 = |em({R})〉. This is due to the parametric dependence of the


monomeric adiabatic states on the monomeric set of intra-molecularnuclear coordinates Rm = (�Rm,1, �Rm,2, . . .), i.e. |mg〉 = |mg(Rm)〉and |me〉 = |me(Rm)〉. Note that the intra-molecular vibrations areseparated from the inter-molecular vibrations. The latter, as well asthe vibrations of the protein environment and their influence on thesystem, can be conveniently described by the bath and the system-bath Hamiltonian. The dependence of the diabatic aggregate stateson the intra-molecular nuclear DOFs can be treated implicitly byincluding the intra-molecular vibrations into the heat bath as well,or explicitly, e.g., by the shifted oscillator model [31]. In the formercase the on-diagonal terms H(g) and Hm(em) are given by the bareelectronic state energies Eg and Eem , respectively. However, in thelatter case the electronic state energies need to be supplemented bya contribution representing the change of the potential energy withrespect to {R} and the corresponding kinetic energy contribution.The corresponding on-site elements of the system Hamiltonian aregiven by

H(g, {R}) = Eg + Ug({R}) + Tg({R}) (2.9)

and

Hm(em, {R}) = Eem + Uem({R}) + Tem({R}). (2.10)

A second order Taylor expansion of the potential energy U({R})around its minimum with respect to {R} in combination with anormal mode transformation yields the frequently used harmonicoscillator model, i.e.

H(g, {q}) = Eg +12∑m

∑ξ

(p

(g) 2ξ,m + ω

(g) 2ξ,m q

(g) 2ξ,m

)(2.11)

and

Hm(em, {q}) =Eem +12

∑n�=m

∑ξ

(p

(g) 2ξ,n + ω

(g) 2ξ,n q

(g) 2ξ,n

)

+ 12∑

ξ

(p

(e) 2ξ,m + ω

(e) 2ξ,m q

(e) 2ξ,m

). (2.12)

2.1. Hamiltonian 9

Here, p(a)ξ,m denotes the momentum and ω

(a)ξ,m the harmonic frequency

associated with the ξth mode of monomer m, being in the electronicstate a, in normal mode space {q}. Note that the normal coordinatesare mass-weighted and that � = 1 here and in the following. It is con-venient to define the normal modes in the corresponding monomericground states |mg〉 and project all vibrational properties of the ag-gregate states |g〉 and |em〉 onto this basis. Assuming additionallythat there are no changes in the curvature of the oscillator potentialsyields the shifted oscillator model. In the limit of the aforementionedapproximations the on-site Hamiltonian is given by

H(g, {q}) = Eg +12∑m

∑ξ

(p2

ξ,m + ω2ξ,mq2

ξ,m

)(2.13)

for the ground state and by

Hm(em, {q}) =Eem +12∑

ξ

(p2

ξ,n + ω2ξ,nq2

ξ,n

)

+ 12∑

ξ

∑n�=m

(p2

ξ,m + ω2ξ,m(qξ,m − dξ,m)2

)(2.14)

for the excited states, respectively. Here, dξ,m represents the shift ofthe excited state oscillator along qξ,m with respect to the ground statepotential energy surface. Introducing intrinsic harmonic oscillatorvariables, i.e.

pξ,m =√

1ωξ,m

pξ,m (2.15)

and

qξ,m =√

ωξ,mqξ,m, (2.16)

yields

H(g, {q}) = Eg +∑m

∑ξ

ωξ,m

2

(p2

ξ,m + q2ξ,m

)(2.17)


and

Hm(em, {q}) =Eem +∑

n�=m

∑ξ

ωξ,n

2

(p2

ξ,n + q2ξ,n

)

+∑

ξ

ωξ,m

2

(p2

ξ,m + (qξ,m − dξ,m)2)

. (2.18)

The dimensionless shift dξ,m can be expressed by the Huang-Rhysfactor Sξ,m ≡ d2

ξ,m/2, which is often used to characterize the couplingbetween electronic and vibrational DOFs. Expanding Eq. (2.18) usingthe definition of Sξ,m yields

Hm(em, {q}) =Eem +∑

ξ

∑n

ωξ,n

2

(p2

ξ,n + q2ξ,n

)

−∑

ξ

ωξ,m

√2Sξ,mqξ,m +

∑ξ

ωξ,mSξ,m. (2.19)

Note that the last sum represents the reorganization energy associatedwith the relaxation within the excited state after a vertical transitionfrom the ground to the excited state and is thus related to the Stokesshift in linear spectroscopy.The interaction between configurations of the aggregate, where the

excitation is located on different sites of the aggregate, i.e. betweendifferent one-exciton states, is in general (in terms of the monomericadiabatic states) given by the Coulomb integral

Jmn(manb, ncmd) =∫d�rd�r ′Nma,md

(�r )Nnb,nc(�r ′)|�r − �r ′| . (2.20)

Here, �r denotes an electronic coordinate and N stands for the gen-eralised molecular charge density [31] depending on the adiabaticmonomeric states. Note that within the present model only elements,which describe a simultaneous excitation of one and de-excitation ofanother monomer, are considered. Thus, the inter-monomeric interac-

2.1. Hamiltonian 11

tions between two sites m and n in Eq. (2.4) are represented by thecoupling elements Jmn(em, en), which are given by

Jmn(em, en) =∫d�rd�r ′ Nem(�r )Nen(�r ′)

|�r − �r ′| , (2.21)

where N denotes the generalized charge density associated with thecorresponding one-exciton state.The Coulomb integral, Eq. (2.21), can be evaluated, e.g., by employ-

ing a transition dipole approximation, assuming that the separationof the monomers is large compared to the spatial extension of thetransition density [31]. However, more elaborate methods, such as thetransition density cube [5], the transition charge from electrostaticpotential method [42], or time-dependent tight-binding-based den-sity functional theory [43], facilitate the calculation of more accurateCoulomb coupling matrix elements.The diabatic one-exciton aggregate Hamiltonian, Eq. (2.6), can

be transformed to an adiabatic representation via diagonalization ofits potential energy part. This leads to a set of adiabatic (delocal-ized) states |α〉 which are connected to the diabatic states via thecomponents cα,m of the eigenvectors of the diabatic Hamiltonian, i.e.

|α〉 =∑m

cα,m |em〉 . (2.22)

For example, for an electronic dimer system the energy of the adiabaticexcited states is given by [31]

E1,2 =Ee1 + Ee2

2 ∓ 12√(Ee1 − Ee2)2 + 4J2

12. (2.23)

Note that the adiabatic levels are numbered according to their energyin increasing order, i.e. the level with the lowest energy is labelled with1, the second lowest with 2 and so forth. The aggregate ground state|g〉 is the same in both representations as there exists no coupling tothe one-exciton states.


The last term in Eq. (2.4) describes interactions with external fields.In dipole approximation Hint is given by

Hint = −dE(t), (2.24)

for the interaction with an external laser field assuming that the field�E(t) and the transition dipole moment �d exhibit a constant orientationwith respect to each other, i.e.

�d · �E(t) = c · |d| |E(t)| . (2.25)

Here, c denotes an orientation factor, which is assumed to be includedin the definition of d in Eq. (2.24). The aggregate dipole operatorwith the site-dependent dipole transition strength dem,g is given by

d =∑m

dem,g|em 〉〈 g|+ h.c.. (2.26)

Under certain conditions all kinds of environments, even those thatcontain anharmonic effects, can be described by an effective harmonicbath [32]. However, the specific form of the bath as well as of thesystem-bath Hamiltonian depends on the underlying model. First,consider the case that only the DOFs of the environment contributeto the bath. This is the case if the intra-molecular nuclear DOFs canbe neglected, e.g., if their coupling to the electronic DOFs is negligibleor if they are included in the system explicitly. The Hamiltonian ofthe environmental DOFs, H

(I)b , is given by

H(I)b =

∑i

ωi

2

(p2

i + x2i

). (2.27)

Here, ωi denotes the harmonic frequency and xi the correspondingbath coordinate. If the intra-molecular vibrations contribute to thebath, it is convenient to separate the bath Hamiltonian into an intra-molecular and an environmental part, i.e.

H(II)b = H

(int)b +H

(env)b . (2.28)

2.1. Hamiltonian 13

The intra-molecular part, H(int)b , is given by, cf. Eq. (2.19),

H(int)b =

∑m

∑ξ

ωξ,m

2

(p2

ξ,m + q2ξ,m

)(2.29)

and the environmental part by Eq. (2.27). Thus, the full bath Hamil-tonian reads

H(II)b =

∑m

∑ξ

ωξ,m

2

(p2

ξ,m + q2ξ,m

)+

∑i

ωi

2

(p2

i + x2i

). (2.30)

The system-bath interaction Hamiltonian, Hs-b, depends in generalon the system and bath operators. However, it can be decomposedinto a product of arbitrary functions of system and bath operators,i.e.

Hs-b = Φb(X)Ks(Q), (2.31)

with no loss of generality. A linear Taylor expansion of the bathfunction Φb(X) and of the system function Ks(Q) yields the widelyused Caldeira-Leggett model [44] for the system-bath Hamiltonian

Hs-b =∑

j

∑ζ

cj,ζXjQζ . (2.32)

Here, Xj denotes an harmonic oscillator position operator and Qζ

an operator representing a part of the system. Note that the linearexpansion with respect to the system operators restricts the descriptionto energy gap fluctuations in the system. An expansion of Ks(Q) upto higher orders of Q would take additional effects into account, e.g.,a modulation of the curvature of the potential energy surface [45].The general form of the Caldeira-Leggett system-bath Hamiltonian,

Eq. (2.32), needs to be adjusted according to the bath and system-bath models (SBMs). Note that the same bath configuration leadsin combination with different SBMs to a different dynamics of thesystem.


a)

SBM Ia

env. bath

SBC SBC

CC

SBM Ib

env. bath

SBC SBC

CC

b)

SBM IIa

env. bathintr.bath 1

intr.bath 2

SBC SBC

CC

SBM IIb

env. bathintr.bath 1

intr.bath 2

SBC SBC

IBC IBC

CC

/c

SBC SBC

Figure 2.1.: Schematic view of the system-bath models (SBM) for adimer system. (a) In SBM Ia/b the DOFs of the two monomers arecoupled to the environmental DOFs by the system-bath coupling (SBC)and to each other via the Coulomb coupling (CC). SBM Ia and b differin the description of the system. In the former the intra-molecularvibrations are included in the system, whereas they are neglected in thelatter. (b) The intra-molecular vibrations are treated as separated bath.In contrast to SBM IIa, where the environmental bath is not coupledto the intra-molecular one, there exists an inter-bath coupling (IBC) inSBM IIb/c.

2.1. Hamiltonian 15

The system-bath Hamiltonian for bath model I, see Eq. (2.27),assuming that the intra-molecular DOFs are included in the systemexplicitly, is given by

H(Ia)s−b =

∑i

∑ξ,m

ci,ξmxiqξ,m|em 〉〈 em|. (2.33)

Here, the index ζ in Eq. (2.32) is associated with the combination ofmode and monomer index (ζ = ξ, m), the general system coordinateQζ equals qξ,m|em 〉〈 em| and the general bath coordinates Xj areidentical with the coordinates of the environmental DOFs, i.e. Xj = xi.If the intra-molecular DOFs can be neglected the expression simplifiesto

H(Ib)s−b =

∑i

∑m

ci,mxi|em 〉〈 em|. (2.34)

Both scenarios are depicted in Fig. 2.1 panel a. Bath model II, seeEq. (2.30) leads to different coupling schemes (SBM IIa-c, Fig. 2.1panel b). However, the discussion will be restricted to SBM IIa and bin the following as the system-bath Hamiltonian for SBM IIc can beconstructed analogously. SBM IIa refers to the situation where thesystem couples separately to the baths of intra-molecular DOFs andto the bath(s) of environmental DOFs, but that the former are notcoupled to the latter. The corresponding system-bath Hamiltonian isgiven by

H(IIa)s−b =

∑m

∑ξ

ωξ,m

√2Sξ,mqξ,m|em 〉〈 em|

+∑m

∑i

ci,mxi|em 〉〈 em|. (2.35)

In contrast, SBM IIb refers to the situation that the system couplesto primary baths of intra-molecular DOFs which couple to secondarybath(s) of environmental DOFs. The system-bath Hamiltonian forSBM IIb is given by

H(IIb)s−b =

∑m

∑ξ

ωξ,m

√2Sξ,mqξ,m|em 〉〈 em|, (2.36)


but note that the bath Hamiltonian needs to be supplemented byan interaction term Hbp−bs between the primary and the secondarybath [46], which is given by

H(IIb)bp−bs =

∑m

∑ξ

∑i

ci,ξmqξ,mxi. (2.37)

Note further that the introduced system-bath models account onlyfor fluctuations of the diagonal elements of the system Hamiltonian.In principle also the off-diagonal Coulomb coupling matrix elementscould be affected by the system-bath interaction [23].

2.2. Equations of motion for the reduced densitymatrix

The system and bath DOFs are still coupled due to system-bathinteraction term Hs-b in Eq. (2.1). However, it is possible to constructequations of motion for the so-called reduced density matrix ρs(t) =trbath {ρ(t)}, which describes the influence of the bath on the systemdynamics implicitly. A perturbative treatment of the system-bathinteraction in combination with the Markov approximation leads to theQME which is widely used to investigate the energy transfer in light-harvesting complexes [11, 21, 22, 31, 47, 48]. QME approaches areonly suitable for weak system-bath interactions as memory effects ofthe bath and higher-order interactions are neglected [49]. In contrast,the HEOM formalism, whose development was initialized by Tanimuraand Kubo in 1989 [34], provides an in principle exact treatment ofthe system-bath interaction.

2.2.1. Quantum master equationThe time evolution of the full density matrix

ρ(t) = U(t, t0)ρ(t0)U †(t, t0) = U(t, t0)ρ(t0) (2.38)

2.2. Equations of motion for the reduced density matrix 17

is determined by the full system Hamiltonian H via the time evolutionoperator

U(t, t0) = e− iH(t−t0) (2.39)

or its Liouville space equivalent U(t, t0). According to the separationof the Hamiltonian, Eq. (2.1), the formal equation of motion of thefull density matrix is given by the Liouville-von Neumann equation

∂

∂tρ(t) =− i [H, ρ(t)]

=− i [Hs, ρ(t)]− i [Hb, ρ(t)]− i [Hs-b, ρ(t)] . (2.40)

In the interaction representation the corresponding equation is givenby

∂

∂tρ(I)(t) = − i

[Hs−b(t), ρ(I)(t)

], (2.41)

where the initial full density matrix in the interaction representationis defined as

ρ(I)(t0) = U †0(t, t0)ρ(t0)U0(t, t0). (2.42)

The time evolution operator U0(t, t0) is defined in analogy to U(t, t0),Eq. (2.39), replacing H by the Hamiltonian of the non-interactingsubsystems Hs +Hb. Inserting the formal integration of Eq. (2.41)into the right-hand side of Eq. (2.41), that is essentially applyingsecond-order perturbation theory with respect to Hs−b, yields

∂

∂tρ(I)(t) =− i

[Hs−b(t), ρ(I)(t0)

]

−t∫

t0

dt′ [Hs−b(t),[Hs−b(t′), ρ(I)(t′)

]]. (2.43)

Assuming further uncorrelated initial conditions for the density matrix,i.e.

ρ(I)(t0) = ρs(t0)ρb(t0), (2.44)


where ρs/b(t0) denotes the equilibrium density matrix of the sys-tem/bath, and that the bath stays in equilibrium independently ofthe amount of energy transferred to it by the system, which is fulfilledif the number of bath DOFs is sufficiently large, the density matrix atan arbitrary time is given by

ρ(I)(t) = ρ(I)s (t)ρb(t0). (2.45)

This leads to∂

∂tρ(I)

s (t) =− trbath { i [Hs−b(t), ρs(t0)ρb(t0)]}

−t∫

t0

dt′ trbath{[

Hs−b(t),[Hs−b(t′), ρ(I)

s (t′)ρb(t0)]]}

.

(2.46)

The integral on the right-hand side of Eq. (2.46) depends on all formertime points due to ρ

(I)s (t′). Therefore, the reduced density matrix

memorizes its own evolution. Expanding the commutators usingEq. (2.31) and taking into account that the system and the bathoperators commute leads to

∂

∂tρ(I)

s (t)

=− iKs(Q(t))ρs(t0) trbath {Φb(X(t))ρb(t0)}+ iρs(t0)Ks(Q(t)) trbath {Φb(X(t))ρb(t0)}− I(Ks(Q(t)), Ks(Q(t′)), ρ(I)

s (t′),Φb(X(t)),Φb(X(t′)))+ I(Ks(Q(t)), ρ(I)

s (t′), Ks(Q(t′)),Φb(X(t′)),Φb(X(t)))+ I(Ks(Q(t′)), ρ(I)

s (t′), Ks(Q(t)),Φb(X(t)),Φb(X(t′)))− I(Ks(Q(t′)), Ks(Q(t)), ρ(I)

s (t′),Φb(X(t′)),Φb(X(t))), (2.47)

where I(A, B, C, D, E) is defined as

I(A, B, C, D, E) =t∫

t0

dt′A · B · C trbath {D · E · ρb(t0)} . (2.48)


Note that the trace over Φb(X(t))ρb(t0) is actually the expectationvalue of Φb(X(t)), which is zero by definition for a harmonic bath, cf.Eq. (2.27). Thus, the first two terms of Eq. (2.47) vanish in this case.Rewriting Eq. (2.47) and introducing the bath correlation function

C(t − t′) = trbath{Φb(X(t))Φb(X(t′))ρb(t0)

}, (2.49)

yields

∂

∂tρ(I)

s (t) =−t∫

t0

dt′Ks(Q(t))Ks(Q(t′))ρ(I)s (t′)C(t − t′)

+t∫

t0

dt′Ks(Q(t))ρ(I)s (t′)Ks(Q(t′))C∗(t − t′)

+t∫

t0

dt′Ks(Q(t′))ρ(I)s (t′)Ks(Q(t))C(t − t′)

−t∫

t0

dt′ρ(I)s (t′)Ks(Q(t′))Ks(Q(t)))C∗(t − t′). (2.50)

Contracting the terms above leads to

∂

∂tρ(I)

s (t) =−t∫

t0

dt′ [Ks(Q(t)), C(t − t′)Ks(Q(t′))ρ(I)s (t′)

]

+t∫

t0

dt′ [Ks(Q(t)), C∗(t − t′)ρ(I)s (t′)Ks(Q(t′))

]. (2.51)

The integral on the right-hand side of Eq. (2.51) cannot be evaluatedanalytically due to the unknown behaviour of ρ

(I)s (t′). However, due

to the system-bath interaction the system will loose the memory of itsevolution in the bath after a characteristic bath correlation time τb.Thus one can approximate ρ

(I)s (t′) by ρ

(I)s (t), assuming that τb is much


shorter than the characteristic time scale of the system dynamics.This approximation is known as the Markov approximation [31]. Fur-thermore, the bath correlation function C(t − t′) can be approximatedby 0 for t − t′ > τb in the framework of the Markov approximationas all correlations are dephasing within τb. Therefore, the upperbound of the integral over t′ can be extended to infinity. Applyingthe Markov approximation to Eq. (2.51) leads to

∂

∂tρ(I)

s (t) =−⎡⎣Ks(Q(t)),

∞∫t0

dt′C(t − t′)Ks(Q(t′))ρ(I)s (t)

⎤⎦

+

⎡⎣Ks(Q(t)), ρ(I)

s (t)∞∫

t0

dt′C∗(t − t′)Ks(Q(t′))

⎤⎦ . (2.52)

Since the bath correlation function C(t− t′) is stationary and dependsonly on the difference of τ = t − t′, one can rewrite the correlationfunction as

C(τ) = trbath {Φb(X(τ))Φb(X(t0))ρb(t0)} . (2.53)

Introducing the operator

Λ =∞∫

t0

dτC(τ)Ks(Q), (2.54)

the corresponding QME in the Heisenberg picture is given by

∂

∂tρs(t) = − i [Hs, ρs(t)]−

[Ks(Q),Λρs(t)− ρs(t)Λ†] . (2.55)

Note again that the approximations which are necessary to deriveEq. (2.55) restrict its applicability to systems with a weak system-bathinteraction and a bath correlation time which is short in comparisonto the time scale of the system dynamics.


2.2.2. Hierarchy equations of motionIn contrast to the QME approach discussed in the previous section,the HEOM formalism [34, 50–54] provides an in principle exact set ofequations of motion for the reduced density matrix in the presenceof arbitrary system-bath interactions. The basic idea of the HEOMapproach is to mimic the system-bath interaction via an infinite hier-archy of auxiliary density matrices (ADM). Due to its computationaldemands the HEOM method is only applicable to rather small systems,composed, e.g., of a few chromophores, in a complex environment.Nevertheless, it can serve as a benchmark method to test the validityof more approximate methods.The HEOM can be derived via path integral calculus utilising the

Feynman-Vernon influence functional formalism [34, 50–52]. Note thatShao and co-workers [53, 54] proposed an alternative method to derivethe HEOM. This approach separates the system and bath degreesof freedom via stochastic fields and leads to the same equations ofmotion as the influence functional formalism.In analogy to Eq. (2.38), assuming again an initial factorization

of ρ(t0) into system and bath parts, cf. Eq. (2.44), the formal timeevolution of the reduced density matrix ρs(t) is given by

ρs(t) = trbath{

U(t, t0)ρ(t0)U †(t, t0)}= U(t, t0)ρs(t0). (2.56)

The time propagator in Liouville space U(t, t0) still depends on thesystem as well as on the system-bath interaction part of the fullsystem Hamiltonian. However, a complete separation of the systemand bath DOFs can be achieved by virtue of the Feynman-Vernoninfluence functional formalism [55] rewriting U(t, t0) in path integralrepresentation. The time evolution operator U(t, t0) in path integralrepresentation is defined as [31]

U(t, t0) =αt∫

α0

Dα e iS[α]. (2.57)


Here, α denotes arbitrary paths in phase space with fixed startingand end points α0 and αt, respectively. The action functional S[α]describes the evolution of the full system. According to Eq. (2.1) it canbe decomposed into a system, a bath, and a system-bath interactionpart. Thus, the time evolution of the full density matrix in pathintegral representation is given by [52]

ρ(αt, α′t, t) =

αt∫α0

Dα

α′t∫

α′0

Dα′ e i(Ss[α]+Sb[α]+Ss−b[α])ρ(α0, α′0, t0)

× e− i(Ss[α′]+Sb[α′]+Ss−b[α′])

= U(αt, α′t, t;α0, α′

0, t0)ρ(α0, α′0, t0). (2.58)

Taking the trace over the bath DOFs leads to [51]

U(αt, α′t, t;α0, α′

0, t0) =αt∫

α0

Dα

α′t∫

α′0

Dα′ e iSs[α]F [α, α′] e− iSs[α′], (2.59)

where the system-bath interaction is fully covered by the Feynman-Vernon influence functional F [α, α′] [55]. Note that while working inLiouville space it is necessary to keep track of the order of the termsas for instance the time evolution operator does not commute withthe density operator.Using the Caldeira-Leggett model for the system-bath interaction,

Eq. (2.32), and restricting the description to a single system operatorQ, the influence functional can be expressed as [52]

F [α, α′] = exp

⎛⎝−

t∫t0

dτQL

⎞⎠ , (2.60)

with

Q =Q[α(τ)]− Q[α′(τ)] (2.61)


and

L =Λ[α(τ)]− Λ†[α′(τ)]. (2.62)

Here, the operator

Λ[α(t)] =t∫

t0

dτC(t − τ)Q[α(τ)] (2.63)

characterises the interaction of system and bath DOFs in terms of thebath correlation function C(t−τ), Eq. (2.49), including memory effects.Note that Λ is actually the non-Markovian equivalent of the operatorΛ defined by Eq. (2.54). Usually, C(t − τ) has a rather complex formdepending on the actual system. However, an exponential form of thecorrelation function, i.e.

C(t) =K∑

k=1Ck(t0) e−γkt, (2.64)

is required for the construction of the HEOM to avoid non-hierarchicalterms [52]. In practice, C(t) needs to be parametrized to fulfil this con-straint. For simplicity a mono-exponentially decaying bath correlationfunction, i.e.

C(t) = C(t0) e−γt, (2.65)

will be assumed in the following.Taking the derivative of ρs(t) with respect to time leads formally

to an equation of motion, which describes the time evolution of thesystem DOFs and incorporates the influence of the bath. Therefore,one needs to evaluate the derivative of the time evolution operatorU(αt, α′

t, t;α0, α′0, t0), which in turn contains the derivatives of the

action term and the influence functional F . Whereas the former,describing the evolution of the unperturbed system, is given by

∂

∂te iSs[α] = − iHs e iSs[α], (2.66)


the latter, containing the effects of the system-bath interaction, reads,cf. Eq. (2.60),

∂

∂tF = −QLF . (2.67)

Equation (2.67) provides no closed solution as there exists no ana-lytical expression for L due to the memory effects contained in Λ,cf. the discussion of Eq. (2.51) and Ref. [52]. Applying the Markovapproximation to Eq. (2.67) would reproduce the QME, Eq. (2.55),immediately. However, introducing the so-called auxiliary influencefunctional F1 one obtains the formal non-Markovian solution

∂

∂tF = − i [Q(− iL)]F = −iQF1, (2.68)

which still provides no closed solution due to the aforementionedreasons. Nevertheless, a closed and formally exact solution of theproblem can be obtained by taking the derivatives of F up to nthorder. This leads to an infinite hierarchy of higher auxiliary influencefunctionals Fn. Considering the time dependence of L, the firstderivative of F1, i.e. the second derivative of F , is given by

∂

∂tF1 =− i

(∂

∂tL

)F − iQF2

=− i{C(t0)Q − C∗(t0)Q}F − γF1 − iQF2, (2.69)

and the derivative of Fn, i.e. the (n+ 1)th derivative of F , by∂

∂tFn = − i{C(t0)Q − C∗(t0)Q}Fn−1 − nγFn − iQFn+1. (2.70)

Note that the time derivatives of L in general could give rise to non-hierarchical terms. Due to the special choice of the bath correlationfunction, Eq. (2.65), i.e. the exponential shape of the function, theseterms obey a hierarchical form as well. The leading order of the system-bath coupling of the auxiliary influence functional Fn is 2n [52]. Thus,in addition to memory effects also higher orders of the system-bathcoupling are included in the HEOM formalism.


The obtained infinite hierarchy of influence functionals leads to aninfinite hierarchy of auxiliary reduced density matrices,

ρn(t) = Unρn(t0), (2.71)

where the auxiliary time evolution operator Un is obtained by replacingthe influence functional F in Eq. (2.59) by the corresponding auxiliaryinfluence functional Fn. The time evolution of the reduced densitymatrix and the ADM can be expressed in terms of a system of coupleddifferential equations, which is given by

∂

∂tρn =− ( iL+ nγ) ρn − i [Q, ρn+1]

− in [C(0)Qρn−1 − C∗(0)ρn−1Q] . (2.72)

Here, L• = [Hs, •] denotes the Liouville superoperator. The initialconditions for the reduced density matrices are ρ0(t0) = ρs(t0) andρn>0(t0) = 0. Note that the density matrix ρ0 represents the reducedsystem density matrix ρs. The ADM represent the influence of thebath and thus the memory effects, but contain no direct informationon the system.So far C(t) was assumed to be a mono-exponential function, which

is rather unlikely for real systems. However, the only constraintfor the bath-correlation function, which is necessary to avoid non-hierarchical terms, is the aforementioned general exponential formof C(t). Following the presented construction algorithm, using thegeneral expansion of the correlation function, Eq. (2.64), leads to thegeneralized HEOM [52]

∂

∂tρn = − iLρn − γnρn + ρ

(+)n + ρ

(−)n . (2.73)

The index array n = (n1, ..., nK) characterizes the leading order ofsystem-bath coupling 2nk of each component k of the correlationfunction. The contributions ρ

(+)n and ρ

(−)n are given by

ρ(+)n = − i

K∑k=1

[Qk, ρn+

k

](2.74)


and

ρ(−)n = − i

K∑k=1

nk

(ckQkρn−

k− c∗

kρn−k

Qk

). (2.75)

Here, the index array n±k = (n1, n2, ..., nk ± 1, ..., nK) denotes the

change of the kth component of the index array n. Note that thecomponents k = 1, . . . , K of the correlation function do not necessarilyneed to be associated with the same interaction. Instead, they might,e.g., account for the interaction of the same environment with differentsubsystems or for the interaction of different environments with thesame subsystem. In the former case the system part of the system-bathHamiltonian Qζ , cf. Eq. (2.32), differs for the different subsystems,whereas in the latter case the correlation functions for the differentenvironments differ. For instance, let us assume a molecular dimersystem, in which the system operators are defined as Qζ=1 = |e1 〉〈 e1|and Qζ=2 = |e2 〉〈 e2| and each monomer is interacting with the sameenvironment. The correlation function shall be approximated bytwo exponential terms with the prefactors C1 and C2. Therefore,the overall correlation function consists of four terms (K = 4). Thecorresponding HEOM coefficients ck and operators Qk are given by

c1 = C1 Q1 = |e1 〉〈 e1|c2 = C2 Q2 = |e1 〉〈 e1|c3 = C1 Q3 = |e2 〉〈 e2|c4 = C2 Q4 = |e2 〉〈 e2|.

According to the index array n, the prefactor of the direct influenceterm is given by

γn =K∑

k=1nkγk. (2.76)

The hierarchy size S is defined by two dimensions, namely the numberof expansion terms of the correlation function and the number of


layers, where the Nth hierarchy layer consists of all density matriceswith the same order of the system-bath coupling, i.e.

N =K∑k

nk. (2.77)

The first three layers of a hierarchy with K = 2 are shown in Fig. 2.2.According to the equations of motion, Eq. (2.73), the propagation ofthe reduced density matrix ρs(t) = ρ00(t) requires the auxiliary densitymatrices ρ10(t) and ρ01(t), which form the first hierarchy layer. Thepropagation of the latter matrices requires beside ρ00(t) additionallythe second layer matrices and so on. Thus, the formally exact solutionwould require an infinite number of ADM and the hierarchy needs tobe truncated for practical purposes.

Layer (N)

0

1

2

ρ00

ρ10 ρ01

ρ20 ρ11 ρ02

Figure 2.2.: Hierarchical scheme of the density matrices in the HEOMformalism for K = 2.

The simplest approach to truncate the hierarchy is to limit thein principle infinite number of correlation function terms to a finitenumber K and the number of layers to a finite number N . The overall


number of reduced density matrices in the truncated hierarchy is thengiven by [35]

S(N , K) =N∑

N=0

(N +K − 1)!N !(K − 1)! . (2.78)

This truncation method is rather inefficient as usually large K andN are required to obtain converged results and the hierarchy sizegrows undesirably fast with increasing K and N . More sophisticatedtruncation schemes have been proposed for both hierarchy dimen-sions [35, 36, 50, 56], which improve the convergence properties of theHEOM significantly. Applying the Markov approximation to the low-est hierarchy layer, Tanimura and co-workers developed a truncationmethod which takes into account the effects of the (N + 1)th layertoo [50]. A similar approach leads to a QME-like approximation ofthe residual part of the bath-correlation function [56]. Furthermore,introducing a normalized set of ADM, an efficient on-the-fly scalingalgorithm has been developed [35], which reduces the numerical effort.This algorithm neglects all ADM whose largest element is below acertain threshold. An improved version of the HEOM, taking intoaccount the residual approximation of the correlation function andthe rescaling of the density matrices, is given by [36]

∂

∂tρn = − iLρn − δRρn − γnρn + ρ

(−)n + ρ

(+)n , (2.79)

with

ρ(+)n = − i

K∑k=1

√(nk + 1)

√|ckck′ |

[Qk, ρn+

k

], (2.80)

ρ(−)n = − i

K∑k=1

√nk√|ckck′ |

(ckQkρn−

k− c∗

k′ρn−k

Qk

)(2.81)

and

δR• =∑

ζ

Δζ [Qζ , [Qζ , •]] . (2.82)

2.3. Spectral density models 29

The index k′ is defined such that k′ = l if γl = γ∗k and Qk = Ql [36],

which means that components belonging to a complex conjugated pairof bath correlation frequencies γk, cf. Eq. (2.64), are grouped. Thereexists always such a pair of conjugated frequencies for oscillatory com-ponents of the correlation function. For non-oscillatory componentsγk is real, thus k′ = k. In contrast to the other summations, which areperformed with respect to k, Eq. (2.80) and Eq. (2.81), the summationwhich accounts for the influence of the residual part of the correlationfunction, Eq. (2.82), is performed with respect to the physical systemoperator index ζ, cf. Eq. (2.32), to avoid a multiple counting of theresidual contributions. The constant Δζ can be calculated using thecoefficients of the residual part of the correlation function, cf. App. A.In addition to the conceptional developments, improved computa-

tional algorithms made it possible to study rather large systems likethe B850 ring system of the light-harvesting antenna complex 2 ofpurple bacteria [37, 57, 58]. These algorithms are based on parallelizedintegration routines and the computation power of graphic processingunits (GPUs).

2.3. Spectral density modelsIn the context of the QME and HEOM formalisms, which were dis-cussed in the previous section, the key property, which determinesthe influence of the bath on the system dynamics, is the bath corre-lation function C(t), Eq. (2.53). Unfortunately, in most cases C(t)cannot be calculated directly as the evaluation of the right-hand sideof Eq. (2.53) would require a quantum dynamics simulation of thefull system. Such simulations are limited to rather small systemsconsisting of some hundred particles.However, the Fourier transform of C(t),

C(ω) =∞∫

−∞dt e iωtC(t), (2.83)


contains all necessary information on the bath characteristics as welland can be determined approximately, e.g., by classical molecular dy-namics simulations or experiments [31, 46]. The spectral distributionfunction C(ω) satisfies the detailed balance condition

C(−ω) = e−βωC(ω). (2.84)

Here, β ≡ (kBT )−1 denotes the inverse temperature. Introducing thesymmetric and antisymmetric contributions of the spectral distributionfunction

C(ω) = C(±)(ω)1± e−βω

(2.85)

and the Bose-Einstein distribution function

n(ω) = 1eβω − 1 , (2.86)

the spectral distribution function can be expressed solely via itsantisymmetric contribution, i.e.

C(ω) = (1 + n(ω))C(−)(ω). (2.87)

Thus, the bath correlation function is given by

C(t) = 12π

∞∫−∞

dω e− iωt(1 + n(ω))C(−)(ω). (2.88)

Expanding the integral on the right-hand side of Eq. (2.88), usingthat C(−) is antisymmetric and

n(−ω) = − eβωn(ω), (2.89)

allows one to express C(t) by the half-sided Fourier integral

C(t) = 12π

∞∫0

dω[e− iωt(1 + n(ω)) + e iωtn(ω)

]C(−)(ω)

= 12π

∞∫0

dω[cos(ωt) coth

(βω

2

)− i sin(ωt)

]C(−)(ω). (2.90)


However, the bath correlation functions C(t) for the system-bathmodels introduced in Sec. 2.1 can be evaluated analytically. The bathcorrelation function for SBM I and IIa can be evaluated introducingcreation and annihilation operators for the harmonic oscillator [31].Due to the coupling of the different bath DOFs to the system DOFsin SBM IIb/c the calculation of the corresponding correlation func-tions requires more sophisticated methods, e.g., path integral tech-niques [46]. Note that the following discussion is therefore restrictedto SBM I and IIa, whereas SBM IIb/c will be discussed separatelylater on.The creation and annihilation operators, whose action on a complete

set of harmonic oscillator eigenstates |Nj〉 is defined by

a†j |Nj〉 =

√Nj + 1 |Nj + 1〉 (2.91)

and

aj |Nj〉 =√

Nj |Nj − 1〉 , (2.92)

are given by

a†j(t) =

1√2(Xj(t)− iPj(t)) (2.93)

and

aj(t) =1√2(Xj(t) + iPj(t)) . (2.94)

Using the operators defined above, the position operator can beexpressed as

Xj(t) =1√2

(aj(t) + a†

j(t))

. (2.95)

Here, again Xj denotes a harmonic oscillator position operator andPj the corresponding momentum operator. Inserting the bath part of


the generic Caldeira-Leggett Hamiltonian, Eq. (2.32), which is givenby

Φb(X) =∑

j

cζ,jXj , (2.96)

in Eq. (2.49), and using Eq. (2.95), yields

Cζ(t) =∑

j

c2ζ,j trbath {Xj(t)Xj(0)ρb(t0)}

=∑

j

12c2

ζ,j trbath{(

aj(t) + a†j(t)

) (aj(0) + a†

j(0))

ρb(t0)}

.

(2.97)Note that the description is restricted to fluctuations of the systemcoordinates, i.e. the overall correlation function separates into com-ponents associated with the system coordinates Qζ . For the presentscenario the overall correlation function is given by

C(t) =∑

ζ

Cζ(t). (2.98)

In general also terms of the type Cζζ′ exist, which describe the fluctu-ations of the correlations between the system coordinates. However,these terms are neglected in the present work.The trace on the right-hand site of Eq. (2.97) can be evaluated

assuming that ρb(t0) describes an equilibrium state which correspondsto a thermal (Boltzmann) distribution of the bath harmonic oscilla-tors, i.e. ρb(t0) = Z−1exp(−βHb), where Z denotes the partitionfunction. Taking into account the time dependence of the creationand annihilation operators leads to

Cζ(t) =∑

j

12c2

ζ,jZ−1 ∑N

〈Nj | e− iωjtaj

(aj + a†

j

)e−βHb |Nj〉

+∑

j

12c2

ζ,jZ−1 ∑N

〈Nj | e iωjta†j

(aj + a†

j

)e−βHb |Nj〉

=∑

j

12c2

j,ζ

∑N

fNj

((1 +Nj) e− iωjt +Nj e iωjt

)(2.99)


with fNj = Z−1exp(−βENj ). The sum over fNj Nj is just the defini-tion of the Bose-Einstein distribution function [31], Eq. (2.86), andthus the bath correlation function is given by

Cζ(t) =∑

j

12c2

ζ,j

[(1 + n(ωj)) e− iωjt + n(ωj) e iωjt

](2.100)

and the corresponding spectral distribution function by

Cζ(ω) =∑

j

πc2ζ,j [(1 + n(ωj))δ(ω − ωj) + n(ωj)δ(ω + ωj)] . (2.101)

Introducing the spectral density

Jζ(ω) =π

2∑

j

c2ζ,jδ(ω − ωj), (2.102)

using the properties of the δ-distribution and Eq. (2.89), the spectraldistribution function Eq. (2.101) can be expressed via the spectraldensity as

Cζ(ω) = 2[1 + n(ω)][Jζ(ω)− Jζ(−ω)]. (2.103)

Note that in analogy to Eq. (2.98) the overall spectral density of thesystem reads

J(ω) =∑

ζ

Jζ(ω). (2.104)

The corresponding antisymmetric contribution of Cζ(ω) is given by

C(−)ζ (ω) = 2[Jζ(ω)− Jζ(−ω)]. (2.105)

Inserting Eq. (2.105) into Eq. (2.90) leads to

Cζ(t) = 〈Φb(X(t))Φb(X(t0))〉

=1π

∞∫0

dω[cos(ωt) coth

(βω

2

)− i sin(ωt)

]Jζ(ω). (2.106)


This equation provides the necessary link between the spectral den-sity function J(ω), which can be obtained by various theoretical andexperimental techniques, with the bath correlation function C(t),needed for quantum dynamics simulations. Note that J(ω) is de-fined only for positive values of ω, thus it is positive by definition.Approximations for J(ω) of natural systems are provided, e.g., byfluorescence line-narrowing and absorption experiments [9, 13] or acombination of molecular dynamics and electronic structure calcula-tions [23, 59]. There exists also a number of model spectral densities,which are frequently used if no experimental or simulation data areavailable [60].The bath-correlation function for a specific system-bath model

can be evaluated by inserting the corresponding bath part of Hs−binstead of the generic Caldeira-Leggett one in Eq. (2.49) and followingthe presented algorithm. This leads for SBM Ib (ζ = m, j = i,cf. Eq. (2.34)) to the monomeric spectral density

J (Ib)m (ω) = π

2∑

i

c2i,mδ(ω − ωi) (2.107)

and for SBM IIa, cf. Eq. (2.35), to

J (IIa)m (ω) =π

∑ξ

ω2ξ,mSξ,mδ(ω − ωξ,m)

+ π

2∑

i

c2i,mδ(ω − ωi). (2.108)

The aforementioned spectral densities are defined in terms of sumsof delta functions weighted by some coefficients. However, any macro-scopic system will in practice have a continuous spectral density dueto the, at least for the environmental contribution, dense spectrumof DOFs. The environmental part of the spectral densities can bedescribed by so-called Ohmic spectral densities, which are charac-terised by a linear rise of J(ω) for small frequencies and an exponentialor Lorentzian cut-off. For example, the monomeric Ohmic spectraldensity with exponential cut-off is given by

Jm(ω) = Θ(ω)ηmω e−ω

γm , (2.109)


where the Heaviside function Θ assures that Jm(ω) = 0 for ω < 0.ηm represents the strength of the system-bath interaction and thecut-off term exp(−ω/γm) determines the shape of Jm(ω) for values ofω which are similar to or larger than the cut-off frequency γm. Themonomeric Ohmic spectral densitiy with a Lorentzian cut-off,

Jm(ω) = Θ(ω)ηmωγ2

m

ω2 + γ2m

, (2.110)

is usually called Debye spectral densitiy. The monomeric correlationfunction associated with the Debye spectral density can be calcu-lated analytically using Eq. (2.106) and the residue theorem. Oneobtains [46]

Cm(t) =ηmγ2

m

2

(cot

(βγm

2

)− i

)e−γmt

+ 2ηmγ2m

β

∞∑ν=1

γν e−γνt

γ2ν − γ2

m

. (2.111)

Here, the summation over the Matsubara frequencies, γν = 2πβ−1ν,stems from the poles of the Bose-Einstein distribution function, i.e. thehyperbolic cotangent in Eq. (2.106), whereas the first term stems fromthe pole of the spectral density itself [46]. Note that the Ohmic andDebye spectral densities describe the environmental part of the overallspectral density fairly well, but provide a rather crude approximationfor the intra-molecular part as the latter is usually not continuous.Comparing the intra-molecular monomeric spectral density,

J (int.)m (ω) = π

∑ξ

ω2ξ,mSξ,mδ(ω − ωξ,m), (2.112)

and the shifted oscillator Hamiltonian, Eq. (2.19), one notes furtherthat the reorganization energy, associated with the relaxation of the


intra-molecular modes after a vertical transition from the monomericground to the excited state, can be evaluated by the integral

1π

∞∫0

dω J(int.)m (ω)

ω=1

π

∞∫0

dω∑

ξ

ω2ξ,mSξ,m

ωξ,mδ(ω − ωξ,m)

=∑

ξ

Sξ,mωξ,m. (2.113)

Accordingly, the corresponding sum of the monomeric Huang-Rhysfactors is given by

1π

∞∫0

dω Jξ,m(ω)ω2 =

∑ξ

Sξ,m. (2.114)

These properties of the spectral densities can be generalized, i.e. theoverall reorganization energy is given by

Ereorg ≡ 1π

∞∫0

dω J(ω)ω

, (2.115)

and the overall coupling strength of the system to the bath by

S ≡ 1π

∞∫0

dω J(ω)ω2 . (2.116)

So far a possible coupling between the different bath DOFs likein SBM IIb/c was neglected. The evaluation of the correspondingcorrelation functions requires, as mentioned above, the use of, e.g.,path integral techniques. For instance, the monomeric spectral densityfor SBM IIb, which corresponds to the frequently used Multi-modeBrownian oscillator (MBO) model, is given by [46]

J (IIb)m (ω) =

∑ξ

Sξ,mω2ξ,m

ωωξ,mγξ,m(ω)(ω2 − ω2

ξ,m)2 + ω2γ2ξ,m(ω)

. (2.117)


Here, the function γ(ω), which accounts for the interaction betweenthe bath DOFs, is given by

γξ,m(ω) =π

2∑

i

c2i,ξm δ(ω − ωi). (2.118)

The HEOM formalism, Sec. 2.2.2, requires a parametrization ofthe correlation function into a series of exponentials. Note that thisparametrization requires a constant γξ,m(ω), i.e. γξ,m(ω) needs to beapproximated by a characteristic frequency γξ,m. This approximationis rather severe as it leads to the non-physical picture that there existsa practically dense spectrum of bath modes, which couple equallystrong to arbitrary intra-molecular modes.Assuming a constant γξ,m(ω), inserting Eq. (2.117) into Eq. (2.106)

and applying the residue theorem, leads to

C(IIb)m (t) =

∑ξ

Ξξ,m

(− e−Ω(+)

ξ,mtΥ(+)

ξ,m + e−Ω(−)

ξ,mtΥ(−)

ξ,m

)

−∑

ξ

Ψξ,m

∞∑ν=1Γ(ν)

ξ,m. (2.119)

The parameters Ξξ,m, Υ(±)ξ,m, Ψξ,m and Γ

(ν)ξ,m are defined as

Ξξ,m =Sξ,mω3

ξ,m

2Θξ,m, (2.120)

Υ(±)ξ,m =coth

⎛⎝ iβΩ(±)

ξ,m

2

⎞⎠ − 1, (2.121)

Ψξ,m =4Sξ,mω3

ξ,mγξ,m

β, (2.122)

and

Γ(ν)ξ,m =

γn e−γnt

(γ2n − ω2

ξ,m)2 + γ2nγ2

ξ,m

(2.123)


with

Ω(±)ξ,m =

γξ,m

2 ± iΘξ,m (2.124)

and

Θξ,m =

√ω2

ξ,m − γ2ξ,m

4 . (2.125)

The first two terms of Eq. (2.119) arise directly from the two polesof the spectral density, Eq. (2.117), whereas the third term arisesfrom the poles of the Bose-Einstein distribution. These poles are asin Eq. (2.111) treated by the Matsubara scheme. Note that thereexist other sum-over-poles schemes, like the Padé scheme, which havebetter convergence properties with respect to the number of termsrequired to model the correlation function with reasonable accuracythan the Matsubara scheme [36].The correlation functions of the Debye and the MBO model have

by construction the exponential form, Eq. (2.64), required by theHEOM formalism, cf. section 2.2.2. This is not the case for arbitraryspectral densities. Meier and Tannor [61] developed a numericalparametrization scheme which is based on the assumption that theactual dynamics of the system only depends on the value of the spectraldensity at the transition energies of the system and not on the shape ofits individual components. They successfully parametrized an Ohmicspectral density, Eq. (2.109), by an expansion into three Lorentzianterms, using the parametrized spectral density,

J(ω) =K∑

k=1xk

ω

[(ω + yk)2 + z2k][(ω − yk)2 + z2

k]. (2.126)

Here, the set of parameters {xk}, {yk} and {zk} for any given spectraldensity J(ω) can be obtained via minimizing the functional

F [{xk}, {yk}, {zk}, K] =∞∫

0

dω|J(ω)− J(ω)|. (2.127)

2.4. Calculation of optical spectra - response function formalism 39

The correlation function according to the parametrized spectral densityJ(ω) is given by

C(t) =K∑

k=1

xk

ykzk[coth(β2 (yk + izk))− 1] e(iyk−zk)t

+K∑

k=1

xk

ykzk[coth(β2 (yk − izk))− 1] e(−iyk−zk)t

+ 2 iβ

∞∑ν=1

J(iγν) e−γνt. (2.128)

2.4. Calculation of optical spectra - responsefunction formalism

The interaction between electromagnetic radiation and molecular sys-tems provides valuable insights into their stationary and dynamicalproperties. There exists a variety of linear and nonlinear spectroscopytechniques, which are suitable to study, e.g., the energy transport inmolecular aggregates. The key property, which connects the macro-scopic response of the system to the applied electric field E(t) withthe underlying microscopic dynamics, is the polarization P [E(t), t].In dipole approximation, i.e. the interaction of the system and thefield is described by Eq. (2.24), P [E(t), t] of a homogeneous sample isgiven by [31]

P [E(t), t] = nmol tr {ρ(t)d} . (2.129)

Here, d denotes the microscopic dipole operator, Eq. (2.26), andnmol stands for the volume density of the molecules in the samplevolume. Note that E(t), d and P [E(t), t] are in general vectors, butfor simplicity the vector character will be neglected in the following.Whereas the microscopic dynamics of the system is described, e.g.,

by the Schrödinger equation or the equations of motion introduced


in Sec. 2.2, the macroscopic connection between the field and thepolarization is determined by the Maxwell equations, i.e.(

∂2

∂t2 − c2Δ)E(t) = −4π ∂2

∂t2 P [E(t), t]. (2.130)

Here, c denotes the speed of light and Δ the Laplace operator. Inprinciple, one needs to solve the coupled equations, given by the fieldequation and the system’s equations of motion, self-consistently todescribe the light-matter interaction exactly. However, often sucha rigorous treatment is not necessary in the limit of weak system-field interaction [46], i.e. it is sufficient to solve first the equations ofmotion of the system driven by the external field and to calculate thecorresponding macroscopic signal afterwards.The functional P [E(t), t] is in general a nonlinear functional of the

electric field, but can be decomposed into its linear and nonlinear com-ponents. In the following sections the connection of the componentsof the polarization to linear and nonlinear spectroscopy techniqueswill be discussed.

2.4.1. Linear ResponseLet us assume a linear relationship between the polarization and theelectric field, i.e.

P = χE, (2.131)

and that the electric field is given by

E(t) = E(t) e ikr− iωt + c.c.. (2.132)

Here, E(t) represents the envelope function of the field with wave-vector �k. The absorption coefficient α(ω) describing the decay ofthe field intensity inside the sample according to Beer’s law is givenby [31]

α(ω) = 4πω

ncIm(χ(ω)), (2.133)


where n denotes the index of refraction of the sample. The dielectricsusceptibility χ(ω) depends on the system properties and can bedetermined microscopically via the so-called linear response functionR(1)(t). Assuming a weak field, which is usually the case in theexperiment, it is convenient to expand Eq. (2.129) in powers of theelectric field strength. This leads to [31]

P (t) = i∞∫

0

dτ Θ(τ)nmol tr{

ρeq[d(I)(τ), d(I)(0)

]}︸︷︷︸

R(1)(t)

E(t − τ). (2.134)

Here, Θ(t) denotes again the Heaviside function and the dipole opera-tor is given in the interaction representation, cf. Sec. 2.2.1, with respectto the system-field interaction. It is assumed further that the initialdensity matrix describes the equilibrium of the system, i.e. ρ(t0) = ρeq.Note that the polarization is given by the response function itself inthe so-called impulsive limit, i.e. E(t − τ) = Eδ(t − τ). Applying theconvolution theorem, the Fourier transform of Eq. (2.134) is given by

P (ω) = χ(ω)E(ω) (2.135)

with

χ(ω) =∞∫

−∞dt e iωtR(1)(t). (2.136)

Thus, the absorption coefficient is proportional to the Fourier trans-form of the linear response function, R(1)(t), i.e.

α(ω) ∝∞∫

0

dt e iωt tr{

ρeq[d(I)(τ), d(I)(0)

]}. (2.137)


Expanding the commutator on the right-hand side of Eq. (2.137) inthe Schrödinger picture leads to

α(ω) ∝∞∫

0

dt e iωt tr{

ρeq(U †(t, t0)dU(t, t0)d − dU †(t, t0)dU(t, t0))}

=∞∫

0

dt e iωt tr{

dU(t, t0)dρeqU †(t, t0)− dU(t, t0)ρeqdU †(t, t0)}

=∞∫

0

dt e iωt tr{

dσ(+)(t)− dσ(−)(t)}

. (2.138)

The time evolution operator U(t, t0) is defined by Eq. (2.39). Note thatthe second term under the trace of the right-hand side of Eq. (2.138)gives rise to anti-resonant contributions and can be neglected [31].Therefore, the absorption coefficient is given by

α(ω) ∝∞∫

0

dt e iωt tr{

dσ(+)(t)}

. (2.139)

So far a homogeneous sample was considered. In the experimentthis condition is not fulfilled as the molecules are in general randomlyoriented and individual molecules are influenced differently by theirenvironment. While the latter effect is partly taken into account bythe system-bath model outlined in the previous sections, the formereffect needs to be incorporated by averaging the spectra over thedifferent orientations of the molecules with respect to the incomingfield [62].

2.4.2. Nonlinear ResponseIn the previous section only the linear contribution of the polarizationwas considered. Expanding the polarization in powers of the fieldstrength, i.e.

P = P (1) + P (2) + . . . (2.140)


and extending the perturbative treatment of the expectation value ofthe polarization, the nth order component of the polarization is givenby [46]

P (n) =∞∫

0

dtn . . .dt1R(n)(tn, . . . , t1)

× E(t − tn) . . .E(t − tn − . . . − t1). (2.141)

The nth order response function is defined as

R(n)(tn, . . . , t1) = innmolΘ(t1) . . .Θ(tn) tr{

ρeqK(n)

}(2.142)

with

K(n) =[[[

. . .[d(I)(tn + . . .+ t1), d(I)(tn−1 + . . .+ t1)

]. . .

],

× d(I)(t1)]

, d(I)(0)]

. (2.143)

In analogy to the linear response function R(1) the nth order one,R(n), contains all necessary information to evaluate the correspondingsignals. The electric field, E(t), is in general defined by Eq. (2.132).However, according to the multi-pulse schemes commonly used inthe experiments, it is convenient to separate the overall field intoindividual components corresponding to the different pulses, i.e.

E(t) =n∑

j=1

(Ej(t) e ikjr− iωjt + E∗

j (t) e− ikjr+ iωjt)

. (2.144)

Note that the wave-vector and the carrier frequency of the signal field,resulting from the nonlinear polarization, need to obey the conditions

�ks = ±j1�k1 ± j2�k2 ± . . . ± jn�kn (2.145)

and

ωs = ±j1ω1 ± j2ω2 ± . . . ± jnωn, (2.146)


due to the conservation of energy and momentum. The integer num-bers ji account for the possibility of multiple interactions of the samplewith the same pulse. The phase-matching condition, Eq. (2.145), lim-its the observable signal in the experiment to certain contributions ofthe overall nonlinear signal. This gives rise to a variety of differentnonlinear spectroscopy techniques focussing on special contributions.Frequently, third order spectroscopy techniques are used to investi-

gate the excitation energy transport in molecular aggregates and otherprocesses. Such nonlinear spectroscopy methods are advantageousin comparison to linear spectroscopy techniques as they probe, dueto the multiple interactions of the system with the field, directlythe dynamics of the system. Beside the commonly used transientabsorption scheme [63], the electronic 2D-spectroscopy technique [64,65] provides a powerful tool to study the dynamics of rather complexsystems [16, 66, 67]. In the former method two laser pulses, which aredelayed with respect to each other, are applied to the sample, whereasin the latter a three-pulse scheme is used. The third-order responsefunction, which describes both techniques, is given by

R(3)(t3, t2, t1) = i3nmolΘ(t1)Θ(t2)Θ(t3)

× tr{

ρeq[[[

d(I)(t3 + t2 + t1), d(I)(t2 + t1)], d(I)(t1)

], d(I)(0)

]}.

(2.147)

Expanding the commutators on the right-hand side of Eq. (2.147)leads to

R(3)(t3, t2, t1) = i3nmolΘ(t1)Θ(t2)Θ(t3)

×4∑

i=1(Ri(t3, t2, t1)− R∗

i (t3, t2, t1)). (2.148)

The individual components Ri, which correspond to different com-binations of interactions, can be associated with physical processes


like ground state bleaching, excited state absorption or stimulatedemission and are given by

R1(t3, t2, t1) = tr{

d(I)(t3 + t2 + t1)d(I)(0)ρeqd(I)(t1)d(I)(t2 + t1)}

(2.149)

R2(t3, t2, t1) = tr{

d(I)(t3 + t2 + t1)d(I)(t1)ρeqd(I)(0)d(I)(t2 + t1)}

(2.150)

R3(t3, t2, t1) = tr{

d(I)(t3 + t2 + t1)d(I)(t2 + t1)ρeqd(I)(0)d(I)(t1)}

(2.151)

R4(t3, t2, t1) = tr{

d(I)(t3 + t2 + t1)d(I)(t2 + t1)d(I)(t1)d(I)(0)ρeq}

.

(2.152)

Note that the terms under the trace are rearranged such that the lastinteraction is the leftmost one. This arrangement can be pictured withthe commonly used double-sided Feynman diagram technique [46],which provides an intuitive picture of the evolution of the densitymatrix. In general the double-sided Feynman diagrams consist oftwo vertical lines representing the bra and the ket of the densitymatrix. Time is evolving from the bottom to the top of the diagramand interactions are represented by diagonal arrows, which have theirorigin at or are pointing to the solid bra or ket lines. Whereas arrowspointing to the bra or the ket lines indicate an excitation process,arrows which have their origin at the bra or the ket lines indicatea de-excitation process. Arrows pointing to the right represent aninteraction with a field contribution of Ej(t)exp( i�kj�r − iωjt) andarrows pointing to the left an interaction with a field contribution ofE∗

j (t)exp(− i�kj�r+ iωjt). There exists a large set of Feynman diagramsdescribing all possible third-order interactions. However, using thephase matching condition, Eq. (2.145), assuming a fixed time orderingof the pulses and neglecting off-resonant terms, i.e. applying therotating wave approximation [46], leads to a small subset of Feynmandiagrams describing all processes, which might be observed using aparticular spectroscopy technique for a given level scheme.


τ

T

t

time

ω1

ω2

ω3

ωs

ω1

ω2

ω3

ωs

ω1

ω2

ω3

ωs

ω1

ω2

ω3

ωs

ω1

ω2

ω3

ωs

ω1

ω2

ω3

ωs

|g> <g|

|g> <α|

|β> <α|

|β> <g|

|g> <g|

|g> <g|

|α> <g|

|α> <β|

|α> <g|

|g> <g|

|g> <g|

|g> <α|

|g> <g|

|β> <g|

|g> <g|

|g> <g|

|α> <g|

|g> <g|

|β> <g|

|g> <g|

|g> <g|

|g> <α|

|β> <α|

|γ> <α|

|α> <α|

|g> <g|

|α> <g|

|α> <β|

|γ> <β|

|β> <β|

SE GSB ESA

R2(3) R1

(3) R3(3) R4

(3) R1*(3) R2

*(3)

Figure 2.3.: Double-sided Feynman diagrams representing the in-teraction pathways leading to the rephasing and nonrephasing 2D-spectroscopy signals.

The aforementioned electronic 2D-spectroscopy technique, whichapplies three independent laser pulses to the sample, offers in prin-ciple eight possible phase-matching directions. However, only twodirections are frequently used in the experiment, namely the photonecho direction �k

(rp)s = −�k1 + �k2 + �k3, which gives rise to the so-called

rephasing Feynman diagrams, and the transient gradient direction�k

(nr)s = �k1 −�k2+�k3, which yields the so-called non-rephasing diagrams.All rephasing Feynman diagrams, Fig. 2.3, have in common that

the first pulse creates a coherence of the type |g 〉〈 α|, whereas thethird pulse creates a coherence of the type |β 〉〈 g| or |γ 〉〈 α|. Here,|g〉 denotes the ground state of the system, |α〉 and |β〉 arbitraryone-exciton states and |γ〉 an arbitrary higher excited state. Duringthe so-called coherence time τ between the first and the second pulse,the system accumulates a phase corresponding to the energy gap


between the states |g〉 and |α〉. The second pulse drives the systeminto a population of the excited state or another coherence. After theso-called population time T between the second and the third pulse,the system is driven into the final coherence by the third pulse. Thesystem accumulates again a phase during the so-called rephasing timet, but now with a different sign counteracting the phase, accumulatedduring the coherence time τ , and thus leading to a rephasing, whichdepends on the system-bath interaction. In non-rephasing Feynmandiagrams the last coherence is of the same type as the first one andtherefore no rephasing occurs. The three rephasing and three non-rephasing Feynman diagrams can also be classified according to theprocess which they represent. Whereas the first two diagrams accountfor stimulated emission (SE) and diagrams three and four for groundstate bleaching (GSB), the last two diagrams represent excited stateabsorption (ESA) which is absent for the aggregate model introducedin Sec. 2.1 as this model is restricted to singly excited states.In the impulsive limit, i.e. Ej(t) ∝ Ejδ(t), the rephasing and non-

rephasing 2D-spectra can be obtained via double Fourier transformof the response function with respect to τ and t [68]. Note that therephasing and non-rephasing spectra include both absorptive anddispersive features. The dispersive features can be eliminated byadding both the non-rephasing and the rephasing spectra [69], thusthe absorptive 2D spectrum is given by [70, 71]

S(ωτ , T, ωt) ∝ Im

⎛⎝ ∞∫

0

dt∞∫

0

dτ[e iωτ τ+ iωttR(3)

nr + e− iωτ τ+ iωttR(3)rp

]⎞⎠(2.153)

with

R(3)rp (t3, t2, t1) = R2(t3, t2, t1) +R3(t3, t2, t1)− R∗

1(t3, t2, t1) (2.154)R(3)

nr (t3, t2, t1) = R1(t3, t2, t1) +R4(t3, t2, t1)− R∗2(t3, t2, t1). (2.155)

A detailed discussion of different features in 2D-spectra arising fromthe underlying level structure is given, e.g., in Refs. [64, 65, 68]. Briefly,


the diagonal ωτ = ωt of the 2D-spectrum for T = 0 fs corresponds tothe linear absorption spectrum, i.e. represents the level structure ofthe sample, whereas off-diagonal peaks indicate a coupling betweendifferent levels. Therefore, a change in the intensity distribution of the2D-spectra for different population times T represents the dynamicsof the system. For example a growth of the off-diagonal features belowthe diagonal indicates an energy transfer or relaxation between thelevels corresponding to the respective diagonal features.The rephasing and non-rephasing response functions can be cal-

culated, analogously to the linear response function, Eq. (2.138), inthe Schrödinger picture. Defining the propagator G(tj , ti) for thefield-free propagation of the density matrix during the interval [ti, tj ],the response functions can be evaluated by

R(3)rp (t3, t2, t1) =

i3 tr{

d(−)G(t3, t2)[d(+), G(t2, t1)

[d(+), G(t1, t0)

[d(−), ρeq

]]]}(2.156)

R(3)nr (t3, t2, t1) =

i3 tr{

d(−)G(t3, t2)[d(+), G(t2, t1)

[d(−), G(t1, t0)

[d(+), ρeq

]]]}.

(2.157)

Here, the operators d(−) and d(+) are defined as

d(−) =∑m

dg,em |g 〉〈 em| (2.158)

d(+) =∑m

dem,g|em 〉〈 g|. (2.159)

Note that the calculation of 2D-spectra is much more demanding fromthe computational point of view, as the propagations need to coverboth the t1 as well as the t3 range.

3. Energy transfer inlight-harvesting complexes

Energy migration within the photosynthetic machineries of algae,bacteria and plants has long been considered as a hopping process,which can be described via rate equations. Further, it was assumedthat the structure of the photosynthetic units is optimized to maximizethe quantum yield of the overall process. However, this simplifiedapproach was questioned after the observation of long lasting coherentoscillations in the 2D-spectra of the FMO complex by Engel andco-workers in 2007 [16]. Since then the effect of coherence on theenergy transport properties and its origin has been one of the majorlines of investigation in the context of photosynthesis [18, 20, 21, 26–28, 72, 73]. The dynamics within light-harvesting antennae after anoptical excitation is rather complex due to their sophisticated structureand the influence of the environment. Therefore, it is instructive tostudy, in addition to natural light-harvesting units, model systems todisentangle the dynamics which lead to the high quantum efficiencyof these complexes.In the following section the dynamics of a generic model dimer will

be investigated using the HEOM method, cf. Sec. 2.2.2, implementedin the Rostock HEOM package, cf. App. A. Here, the issue of coherentoscillations as well as the general influence of the bath on the systemdynamics in dependence on the system-bath interaction strength willbe addressed. Next, the energy transfer in an octamer forming anenergy funnel towards a sink will be studied as a model aggregate whichresembles, e.g., the function of photosynthetic antennae, cf. Fig. 1.1.Finally, the energy transfer of the FMO complex itself will be modelledusing a spectral density extracted from experimental data.


50 3. Energy transfer in light-harvesting complexes

3.1. Dimer systemThe dimer system represents the smallest possible molecular aggregate.Due to its low dimensionality it can be treated numerically ratherexactly in comparison to larger aggregates. Hence the dimer is ofparticular interest as a model system which provides a reference formore complicated systems [20, 22, 24, 26, 27, 57, 71–78].To investigate the dissipative exciton dynamics in dimer systems

a generic model incorporating one intra-molecular mode modelledvia a single mode MBO spectral density, Eq. (2.117), is used in thefollowing. The electronic energy gap between the diabatic excitedaggregate states (ΔE = Ee2 −Ee1 = 500 cm−1), cf. Eq. (2.4), is chosensuch that it corresponds to the frequency of the intra-molecular modeω1,1 = ω1,2 = ωvib = 500 cm−1. This situation is rather typical forlight-harvesting complexes, e.g., in the FMO complex there existsa mode with ωvib = 180 cm−1, which is resonant to the gap energybetween monomer three and four, cf. Sec. 3.3. This facilitates astrong mixing of the electronic and vibrational DOFs, which has beendiscussed as one possible origin of long-living oscillations in the 2D-spectra of FMO [21]. Note that all parameters except of the electronicstate energies are assumed to be the same for both monomers, e.g.,S1,1 = S1,2 = S, and therefore the monomer index m and the modeindex ξ will be skipped. The temperature of the bath is chosen to beT = 300 K.In order to study the influence of the vibronic coupling strength, i.e.

the Huang-Rhys factor S, on the system dynamics, four different dimerscenarios are considered. They are characterised by means of theirlinear absorption spectra, population dynamics and 2D-spectra. Thescenarios cover in particular the cases of weak (I and II) and strong(III and IV) vibronic coupling for two different Coulomb couplings,see Tab. 3.1, Each scenario will be investigated for two different inter-bath coupling strengths, namely γ = 50 cm−1 and γ = 200 cm−1.Example inputs for the Rostock HEOM package are provided inapp. B. Converged results for all scenarios were obtained with K = 2and N = 9, cf. Sec. 2.2.2.

3.1. Dimer system 51

Table 3.1.: Huang-Rhys factors, Coulomb coupling strengths and ratiosof the Coulomb coupling strength and vibrational frequency for thedifferent dimer scenarios.

scenario S J [cm−1] J/ωvib

I 0.05 250 0.5II 0.05 750 1.5III 0.5 250 0.5IV 0.5 750 1.5

The linear absorption spectra for I and II, Fig. 3.1 panels a and b1,show only a significant absorption at the energies correspondingto the electronic dimer, cf. Eq. (2.23), whereas the spectra for IIIand IV, panels c and d, show pronounced additional side peaks dueto the strong vibronic coupling. In all cases the second adiabaticelectronic state and the associated vibronic states carry the mainpart of the oscillator strength, cf. Fig. 3.4. This is characteristicfor so-called H-aggregates and a direct consequence of the Coulombcoupling configuration (J > 0) [31]. Aggregates with J < 0, theso-called J-aggregates, show pronounced peaks at the low-energy siteof the spectra, i.e. the first adiabatic state and the associated vibronicstates carry the main part of the oscillator strength. In contrast to thespectra for I, II and IV, the spectrum for III, where the reorganizationenergy associated with the vibronic coupling is comparable to theCoulomb coupling strength, shows a rather complex shape. Increasingthe inter-bath coupling strength γ leads to a larger line width in allcases. As a consequence of the chosen system-bath model, cf. Sec. 2.1,this effect is again more pronounced for III and IV. The line widthis determined by both, the coupling of the electronic DOFs to theprimary bath, i.e. the Huang-Rhys factor S, and the coupling betweenthe primary and secondary bath, i.e. γ. For I and II the smallHuang-Rhys factor is limiting the interaction between the system

1 Please visit www.springer.com and search for the author’s name to access thechapters colored figures.


(E-E0)/ωvib

abso

rptio

n (n

orm

aliz

ed)

J/ωvib=0.5 S=0.05

a)I

-2 -1 0 1 2 3 4

J/ωvib=0.5 S=0.5

c)III

J/ωvib=1.5 S=0.05

b)II

-2 -1 0 1 2 3 4

J/ωvib=1.5 S=0.5

d)IV

Figure 3.1.: Absorption spectra corresponding to the four dimer sce-narios (γ = 50 cm−1 solid red line, γ = 200 cm−1 dashed blue line)calculated with the Rostock HEOM package using Eq. (2.139). Thespectra are normalized such that the peak value of the dominant peakis equal to unity. The reference energy E0 is equal to the mean of thediabatic electronic state energies (E0 = ΔE/2).

and the bath and therefore restricts the line width independentlyof the magnitude of the inter-bath coupling strength. In contrastfor III and IV the interaction between electronic DOFs and primarybath is strong, i.e. the interaction between primary and secondarybath directly influences the decoherence rate of the system DOFsand therefore the line width of the spectra. Note that peaks with asignificant vibrational contribution, cf. discussion of Fig. 3.4, i.e. thepeaks corresponding to the vibronic progression in the panels c and dof Fig. 3.1, show a stronger line broadening for increasing γ thanpeaks with mainly electronic character. This is due to the strongerinteraction of vibrationally excited states with the bath in comparisonto the vibrational ground state.


0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5

ρ22

t [ps]

I: J/ωvib=0.5 S=0.05 II: J/ωvib=1.5 S=0.05 III: J/ωvib=0.5 S=0.5 IV: J/ωvib=1.5 S=0.5

e)0.0

0.2

0.4

0.6

0.8

1.0

ρe 2

e 2

a)

0.0 0.5 1.0 1.5

t [ps]

f)

b)

0.0 0.5 1.0 1.5

t [ps]

g)

c)

0.0 0.5 1.0 1.5 2.0

t [ps]

h)

d)

Figure 3.2.: Population dynamics of the highest diabatic (upper row)and adiabatic (lower row) states for the four dimer scenarios (γ =50 cm−1: solid red line, γ = 200 cm−1: dashed blue line).

The population dynamics of the diabatic and adiabatic states,Fig. 3.2, after initial excitation of the highest adiabatic state is signifi-cantly influenced by the inter-bath coupling for all scenarios. Whereasfor γ = 50 cm−1 all populations except of those for II (panels b and f)feature pronounced oscillations over at least 500 fs, these oscillationsare damped out rapidly for γ = 200 cm−1. Additionally, the relaxationtowards the first diabatic/adiabatic state is significantly faster for thestronger inter-bath coupling.The population dynamics of the diabatic state |e2〉, as well as

of the adiabatic state |2〉, Eq. (2.22), for I, III and IV shows anexponential decay of the state population. This is in contrast to III,where the diabatic and adiabatic populations decay almost linearly.In general the population relaxation for the scenarios featuring a


0.0 1.0 2.0 3.0

FF

T[ρ

22] (

arb.

u.)

ω/ωvib

e)


FF

T[ρ

e 2e 2

] (ar

b. u

.)

a)

0.0 1.0 2.0 3.0

ω/ωvib

f)

b)

0.0 1.0 2.0 3.0

ω/ωvib

g)

c)

0.0 1.0 2.0 3.0 4.0

ω/ωvib

h)

d)

Figure 3.3.: Fourier spectra of the oscillatory component of the popu-lations shown in Fig. 3.2 (γ = 50 cm−1: solid red line, γ = 200 cm−1:dashed blue line). The Fourier spectra are normalized such that thearea under the curve is equal to unity.

strong vibronic coupling (III and IV) is faster than the one for thescenarios featuring a weak vibronic coupling (I and II). Comparingthe population dynamics for I and II as well as those for III andIV, respectively, one further notices that the population relaxationis faster for the scenarios with a smaller Coulomb coupling. This isdue to the fact that the energy gap between the adiabatic electronicstates, cf. Eq. (2.23), for J = 250 cm−1 is closer to ωvib than forJ = 750 cm−1. Thus, the value of the spectral density, which has itsmaximum at ω = ωvib, at the transition energy between the statesis larger in the former case than in the latter case. This leads toa stronger coupling to the bath for J = 250 cm−1, which in returnfacilitates a faster relaxation for I and III in comparison to II and IV.


Comparing the diabatic and adiabatic population dynamics onenotices that, although the general behaviour is the same for both, theadiabatic populations feature fewer oscillations than the correspondingdiabatic ones. In order to characterise the oscillatory behaviour ofthe population dynamics in more detail, it is instructive to inspectthe Fourier spectrum (FS) of the oscillations. It can be calculated bysubtracting the decaying component, i.e.

Posc(t) = P (t)− Pdec(t), (3.1)and taking the Fourier transform of the Posc(t). The decaying compo-nent Pdec(t) of the population can be extracted, e.g., via an exponentialfit, i.e.

Pdec(t) = P (0) e−tτ . (3.2)

According to the population dynamics shown in Fig. 3.2, the FS ofthe adiabatic populations, Fig. 3.3, show in general less features thanthe FS of the corresponding diabatic ones. In particular for I, II andIV the distinct peaks at the high frequencies in the FS of the diabaticpopulations are absent in the spectra of the adiabatic populations.This indicates that the oscillations corresponding to these peaks havetheir origin in the mixing of local vibronic transitions by the Coulombcoupling.This statement can be illustrated considering the exact population

dynamics of the excited states of an electronic dimer without couplingto a bath. The dynamics of this system in both representations isgiven by the Schrödinger equation

H |Ψ(t)〉 = i ∂

∂t|Ψ(t)〉 , (3.3)

where the Hamiltonian in the diabatic basis is given by

Hdia =(

Ee1 J

J Ee2

)(3.4)

and the corresponding adiabatic Hamiltonian by

Hadia =(

E1 00 E2

). (3.5)


The adiabatic state energies are connected to the diabatic onesvia Eq. (2.23). Expanding the time-dependent wave function |Ψ(t)〉in terms of the diabatic and adiabatic states yields

|Ψ(t)〉dia =(

Ae1(t) e− iEe1 t |e1〉Ae2(t) e− iEe2 t |e2〉

)(3.6)

and

|Ψ(t)〉adia =(

A1(t) e− iE1t |1〉A2(t) e− iE2t |2〉

), (3.7)

respectively. The population of an arbitrary state is given by theabsolute square of the corresponding expansion coefficients, i.e.

Pi = |Ai|2 = AiA∗i . (3.8)

Inserting the wave function and the Hamiltonian in Eq. (3.3) leads toequations of motions for the expansion coefficients Ai. In the adiabaticrepresentation the corresponding equations are given by

∂

∂tAi(t) = 0, (3.9)

which yields immediately that the expansion coefficients and thus thepopulations of the adiabatic states are constant. However, due to theCoulomb coupling the equations of motion of the diabatic expansioncoefficients, i.e.

∂

∂tAe1(t) = − iJ e iωe1e2 tAe2(t) (3.10)

and∂

∂tAe2(t) = − iJ e− iωe1e2 tAe1(t) (3.11)

with ωe1e2 ≡ Ee1 − Ee2 , are coupled to each other. Solving the systemof equations leads to

Ae2(t) = b1 e iΩ1t + b2 e iΩ2t, (3.12)


where the coefficients bi need to be adjusted according to the initialconditions and the frequencies Ωi are defined as

Ω1,2 =ωe1e2

2 ± 12√

ω2e1e2 + 4J2. (3.13)

Thus, the population of the state |e2〉 reads

Pe2(t) =Ae2(t)A∗e2(t)

= |b1|2 + |b2|2 + b1b∗2 e i(Ω1−Ω2)t + b∗

1b2 e− i(Ω1−Ω2)t (3.14)

and the population oscillates with a frequency of Ω1 −Ω2, which corre-sponds to the energy gap between the adiabatic states, cf. Eq. (2.23).The oscillation frequencies of the rightmost peaks in the FS of the

diabatic populations for I and II, Fig. 3.3 panels a and b, coincideindeed very accurately with the gap energy of the adiabatic electronicstates, whereas the corresponding peak for IV is shifted towardshigher frequencies. This is a result of the mixing of the electronic andvibrational DOFs due to the strong vibronic coupling, which leads to ahigher energy of the vibronic levels in comparison to the correspondinglevels in the purely electronic model. In III the electronic peak in theFS of the diabatic population can barely be identified due to the largeeffect of the vibronic coupling, although there are some changes inthe FS of the adiabatic population in comparison to the FS of thediabatic population. To summarize, one peak in the FS of the diabaticpopulations can be assigned to be of electronic origin. This peak isabsent in the FS of the adiabatic populations due to the nature of thetransformation between diabatic and adiabatic representation. Thus,in contrast to the diabatic populations, the adiabatic populations,Fig. 3.2, feature no oscillations of electronic origin.However, the vibronic and the system-bath couplings lead not only

to a relaxation onto energetically lower states, but also to additionaloscillations even for a weak vibronic coupling (I and II) although thiseffect is more pronounced for a strong vibronic coupling (III and IV).The peaks in the FS, which have their origin in the vibronic coupling,show a stronger dependence on the bath coupling strength γ than the


electronic peaks. In particular, a larger peak broadening indicates astronger damping of the corresponding oscillations with increasing γin comparison to their electronic counterparts.In analogy to the electronic dimer, the key property to understand

the population dynamics of the vibronic dimer and especially itsoscillatory behaviour for arbitrary Huang-Rhys factors is the fulladiabatic vibronic level structure, cf. app. C. Similar to the frequencyof the electronic oscillations in the diabatic electronic representation,which corresponds to the energy gap of the adiabatic electronic states,the frequency of the vibronic oscillations in the diabatic vibronicpicture corresponds to the energy gap between certain adiabaticvibronic levels. Due to the implicit treatment of the intra-molecularvibrations, the diabatic populations, displayed in Fig. 3.2, correspondto a diabatic vibronic representation and therefore feature the vibronicoscillations. Note that the electronic adiabatic populations shown inFig. 3.2 are not adiabatic with respect to the vibronic basis and thusstill feature vibronic oscillations, cf. app. C. The individual populationsof the adiabatic and diabatic vibronic states are not available in thecontext of an implicit treatment of the intra-molecular vibrations asa heat bath. Nevertheless, the diabatic vibronic level scheme can bemimicked in the present system-bath model, as the properties of theintra-molecular modes, S and ωvib, are known.Introducing the diabatic vibronic states |abμν〉, where a and b

denote the electronic state and μ and ν the vibrational state of thefirst and second monomer, respectively, allows one to calculate theenergy and oscillator strength. The latter is given by the normalizedBoltzmann-weighted transition dipole square of the adiabatic vibronicstates ∣∣α′⟩ = ∑

abμν

cabμν |abμν〉 , (3.15)

which can be obtained via numerical diagonalization of the diabaticvibronic Hamiltonian [79, 80]. The diabatic vibronic aggregate statesrepresent either a vibronic excitation of the first monomer (|egμ0〉),the second monomer (|ge0ν〉), a vibrational excitation of the vibroni-cally unexcited monomer (|geμν〉 with μ > 0 or |egμν〉 with ν > 0), or


a vibrational excitation of the aggregate ground state (|ggμν〉). Fur-thermore, the character of the adiabatic vibronic aggregate states canbe classified according to the overall number of vibrational excitationsin the excitonically excited aggregate states, i.e.

χ(0) = c2ge00 + c2

eg00 (3.16)χ(1) = c2

ge10 + c2ge01 + c2

eg10 + c2eg01 (3.17)

χ(2) = c2ge20 + c2

ge02 + c2eg20 + c2

eg02 + c2ge11 + c2

eg11 (3.18)...

The contribution of the states with zero vibrational excitationsχ(0) can be identified as the electronic contribution. Inspecting theproperties of the adiabatic vibronic levels, Fig. 3.4, one can see thatthe distribution of the oscillator strength corresponds to the intensitydistribution in the absorption spectra, Fig. 3.1.For the weak vibronic coupling (I and II), levels with a large elec-

tronic contribution carry the major part of the oscillator strength,whereas for the strong vibronic coupling (III and IV) also levels with alarge vibrational contribution gain significant oscillator strength. Thisleads to the appearance of vibronic peaks in the corresponding absorp-tion spectra. Further, the peak broadening characteristics for III andIV, in particular that some peaks show a stronger broadening withincreasing γ than others, can be explained considering the characterof the underlying adiabatic vibronic levels. For example, the peakat (E − E0)/ωvib ≈ 3 in Fig. 3.1 panel d, which corresponds to the17th adiabatic vibronic level (Fig. 3.4 panel d), shows a larger peakbroadening with increasing γ than the peak at (E − E0)/ωvib ≈ 2,which corresponds to the eleventh level. This is correlated to the char-acter of these levels, i.e. the eleventh level has a dominant electroniccontribution, whereas the 17th level has a dominant χ(1). Accordingto the present system-bath model, in which the intra-molecular vi-brations are coupled to the environmental DOFs, adiabatic vibronicaggregate states with a stronger vibrational excitation are strongercoupled to the environmental bath. Thus, states, which have a larger


-2.0-1.00.01.02.03.04.0

0 5 10 15 20 25

(E-E

0)/ω

vib

α’

I: J/ωvib=0.5 S=0.05 II: J/ωvib=1.5 S=0.05

III: J/ωvib=0.5 S=0.5 IV: J/ωvib=1.5 S=0.5

a) b)

c) d)

0.0

0.5

1.0

osc.

str

.

0.0

0.5

1.0

cont

r.

0 5 10 15 20 25

α’

-2.0-1.00.01.02.03.04.0

0 5 10 15 20 25

(E-E

0)/ω

vib

α’

0.0

0.5

1.0

osc.

str

.

0.0

0.5

1.0

cont

r.

0 5 10 15 20 25

α’

Figure 3.4.: Energy, oscillator strength, χ(0) (red squares), χ(1) (bluetriangles) and χ(2) (black circles) of the adiabatic vibronic states corre-sponding to the four dimer scenarios. The properties were obtained viadirect diagonalization of a diabatic vibronic Hamiltonian incorporatingten vibrational levels for the ground and each excited monomeric state,respectively. The reference energy E0 is defined as in Fig. 3.1.

χ(n>0) in comparison to some other states, are stronger affected bythe inter-bath coupling than the latter.


In order to interpret the oscillatory behaviour of the populations, itis necessary to select the contributing diabatic and the correspondingadiabatic vibronic states. Owing to the large number of adiabaticvibronic states and their, especially for strong vibronic couplings,dense energy spectrum, this task requires the implementation ofcertain rules. These can be developed inspecting the diabatic vibronicHamiltonian, Eq. (C.4). First, at least one of the involved diabaticvibronic levels needs to be initially populated. Second, having in mindthe analytical solution of the electronic model one notes that onlycoupled diabatic vibronic levels feature oscillations. The couplingbetween the diabatic vibronic levels is determined by the product ofthe Coulomb coupling strength J and the Franck-Condon overlaps ofthe vibrations in the monomeric ground and excited states, Eq. (C.6).Note that the Franck-Condon overlap depends crucially on the Huang-Rhys factor S. For small Huang-Rhys factors only the Franck-Condonoverlaps of configurations, which differ in none or one vibrationalquantum, have a pronounced amplitude. Therefore, only pairs ofadiabatic vibronic aggregate states, of which one has a significantχ(v) and the other has a significant χ(v) or χ(v±1), couple strongly. Incontrast, the Franck-Condon overlaps of configurations, which differ inmore than one vibrational quantum, are no longer negligible for largerHuang-Rhys factors. Thus, additional couplings become importantand one needs to consider also pairs of states, of which one has asignificant χ(v) and the other has a significant χ(v±n). The value of ndepends crucially on S.Due to the choice of the initial conditions for the investigation of

the population dynamics of the four scenarios, i.e. a population of thesecond adiabatic electronic level, only vibronic states with a significantelectronic contribution are initially populated. Following the rulesintroduced above it is possible to reveal the origin of the peaks in theFS. First, one needs to identify the levels with large oscillator strengths.Second, all energetically lower levels, whose energy gap to one of theidentified levels matches the peak frequency need to be considered.Finally, one needs to compare the χ(v) of the identified levels with theχ(v) and χ(v±n) of the considered energetically lower levels. Via the


first two steps of the presented algorithm the first peak in the FS ofthe populations for I, Fig. 3.3 panels a and e, can be assigned to theenergy difference between the fourth and second, and the fourth andthird level, respectively. Whereas level four has a dominant electroniccontribution the levels two and three are of dominantly vibroniccharacter. Thus, the corresponding oscillations in the populations,cf. Fig. 3.2, are of vibronic origin, which is in agreement to the resultobtained by a comparison between the adiabatic and diabatic FS.The same algorithm works for the first and second peak in the FS forII, which can be assigned to the energy gaps between the eleventhand the sixth, the eleventh and 21st and the eleventh and 22nd level,respectively. For III and IV, the adiabatic vibronic level structureis much more complex leading to a variety of superpositions, whichyield the complex oscillatory behaviour of the populations.The initial condition used so far, i.e. the population of the sec-

ond adiabatic state, is rather different in comparison to an opticalexcitation of the system and thus the dynamics after absorption oflight might differ significantly from the dynamics displayed in Fig. 3.2.The 2D-spectroscopy technique provides a powerful experimental toolto study the population dynamics after an optical excitation andthe correlations between the different states [64, 65, 68]. However,the interpretation of the obtained 2D-spectra suffers from the highcomplexity of the investigated systems. Thus, comparison between ex-perimental and calculated 2D-spectra provides not only a benchmarkfor the used theoretical model, but also facilitates the interpreta-tion of the experimental spectra. The 2D-spectra for the differentdimer scenarios were calculated using the response function formalismpresented in Sec. 2.4 with the Rostock HEOM package.Note that the calculation of 2D-spectra is computationally demand-

ing even for small systems as the response functions, Eq. (2.157),need to be evaluated for various combinations of t1, t2 and t3 toobtain sufficient data. The resolution of the calculated 2D-spectradepends crucially on the length of the time intervals [t(min)

1 ; t(max)1 ]

and [t(min)3 ; t(max)

3 ] covered by the propagation. Additionally the time



(ωτ-E0)/(ωvib)

(ωt-E

0)/(

ωvi

b)

-2-101234

-2 -1 0 1 2 3 4

-1 -0.5 0 0.5 1

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2-101234

T=0.5ps

T=0.5ps

T=0.5ps

T=0.5ps

-2-101234

T=0.2ps

T=0.2ps

T=0.2ps

T=0.2ps

-2-101234

T=0.0ps

D

D O

O

T=0.0ps

D

T=0.0ps

T=0.0ps

Figure 3.5.: 2D-spectra for the four dimer scenarios and weak inter-bath coupling γ = 50 cm−1 for T = 0 fs, T = 200 fs, T = 500 fs andT = 1000 fs. The spectra were normalized according to the maximumof the spectra for T = 0 fs. The diagonal and off-diagonal peaks for Iand II are indicated by D and O in the upper panels, respectively

step between different t1/3, i.e. the sampling of the correspondingintervals, needs to be accurate enough to cover all spectral features.The 2D-spectra for the different dimer scenarios, Fig. 3.5 and

Fig. 3.6, were calculated using a sampling of 0.5 fs and an interval



(ωτ-E0)/(ωvib)

(ωt-E

0)/(

ωvi

b)

-2-101234

-2 -1 0 1 2 3 4

-1 -0.5 0 0.5 1

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2 -1 0 1 2 3 4

T=1.0ps

-2-101234

T=0.5ps

T=0.5ps

T=0.5ps

T=0.5ps

-2-101234

T=0.2ps

T=0.2ps

T=0.2ps

T=0.2ps

-2-101234

T=0.0ps

D

D O

O

T=0.0ps

D

T=0.0ps

T=0.0ps

Figure 3.6.: Same as Fig. 3.5 for γ = 200 cm−1.

length of 2048 fs for both τ and t. According to this setup 40962, i.e.approximately 17 million, calculation steps are needed to calculate one2D-spectrum. The resolution of the 2D-spectra is Δωt/τ ≈ 4 cm−1,which is sufficient for an analysis with respect to possible oscillatoryfeatures of the peaks. However, due to the computational effort nosufficient resolution for T could be obtained during the present work.Furthermore, the response functions do not decay sufficiently during


the interval length of 2048 fs, which leads to instabilities of the Fouriertransform with respect to τ and t, Eq. (2.153). In comparison, thelinear absorption spectra, Fig. 3.1, were obtained with a propagationtime of 16 ps. Due to the instabilities of the Fourier transform,negative features arise in the 2D-spectra, which should be absent asthe excited state absorption is not present in the current model. Theconvergence level of the hierarchy was decreased as well, i.e. N = 5instead of N = 9, to reduce the overall computational effort.The diagonals (ωτ = ωt) of the 2D-spectra for the different scenarios

for T = 0 fs resemble the linear absorption spectra, cf. Fig. 3.1. Similarto the linear absorption spectra, the 2D-spectra for I, II and IV haveless features than the spectra for III. Again, the spectra for I and IIare not so strongly affected by an increase of the inter-bath couplingconstant γ as their peaks correspond mainly to levels with dominantelectronic character. The spectra for III show a variety of distinctcross-peaks, which indicate, similar to the FS of the populations, acoupling of the corresponding aggregate states, whereas the spectrafor I, II and IV reveal only a few weak cross-peaks. Furthermore, thespectra for all scenarios show a rapid intensity redistribution withinthe first 200 fs but reveal in contrast to the population dynamicsno significant changes for T = 500 fs and T = 1000 fs. This isdue to the difference in the initial conditions. For the populationdynamics simulation, Fig. 3.2, the initial condition was chosen suchthat only the second adiabatic electronic level was populated, whereasthe interaction with a field excites the vibronic levels. To achieve asimilar distribution of the initial populations and take into accountthe interactions of the system with the second and third pulse, oneneeds to propagate the system treating the system-field interactionexplicitly, e.g., via the interaction operator Hint, Eq. (2.24). Note thatthis feature is not implemented in the Rostock HEOM package so far.Another difference between the population dynamics and the dynamicsof the spectra arises due to the fact that the lower vibronic states, towhich the population relaxes, carry only a small part of the oscillatorstrength according to the positive Coulomb coupling (H-aggregate).Therefore, the dynamics of these states is not observable.


To summarize this section, the population dynamics of dimer sys-tems depends strongly on the interplay between the system and thebath. In particular the value of the spectral density at the transitionenergy between the adiabatic levels, which corresponds to the couplingstrength of the transition to the bath, is crucial for the relaxationdynamics. Thus in the context of the used MBO model for a par-ticular set of system variables, i.e. the Coulomb coupling strengthJ and the energy of the diabatic states Ee1 and Ee2 , an increase ofthe vibronic coupling strength (the Huang-Rhys factor S) will leadto a faster relaxation. Further, for a constant vibronic coupling anda fixed diabatic state configuration, the Coulomb coupling strengthJ determines the dynamics as it tunes the energy gap between theadiabatic states. The oscillatory features, which appear in the popu-lation dynamics, can be interpreted using the full adiabatic vibroniclevel structure of the system. However, this structure is not availablefor larger aggregates. Nevertheless, as a rule of thumb one can assignoscillations, which appear in the diabatic but not in the adiabaticelectronic population, to be of electronic origin. The exact correlationbetween the population dynamics and the dynamics of the 2D-spectrawill be in the focus of further investigations.

3.2. Funnel-like aggregateIn the previous section the population dynamics of a dimer system witha single component MBO spectral density was investigated. However,light-harvesting complexes usually consist of several molecules in arather complex environment. The specific tasks of these specializedcomplexes in the overall photosynthetic process might vary. Whereassome complexes are specialized on the absorption of light and serve aslight-harvesting antennae, e.g., the LH2 complex of purple bacteria,others, like the FMO complex of green sulphur bacteria, serve as energyfunnels, which collect the energy of the antennae and transfer it to thereaction centre. There the energy is used for charge separation, whichdrives the chemical processes of photosynthesis [1]. To address the

3.2. Funnel-like aggregate 67

question of how the environment influences the population dynamics ina funnel-like aggregate, a model system consisting of eight monomerswill be considered in the following. The monomers are grouped intofour dimers, cf. Fig. 3.7 panel a. Note that such a dimerization iscommon in various light-harvesting complexes, e.g., the LH2 complexof purple bacteria [5, 6] and the FMO complex. The one-exciton partof the system Hamiltonian, cf. Eq. (2.6), is given by

H(1)s = 1Ec +

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

32ΔE J1 0 0 0 0 0 0

J132ΔE J2 0 0 0 0 0

0 J212ΔE J1 0 0 0 0

0 0 J112ΔE J2 0 0 0

0 0 0 J2 −12ΔE J1 0 0

0 0 0 0 J1 −12ΔE J2 0

0 0 0 0 0 J2 −32ΔE J1

0 0 0 0 0 0 J1 −32ΔE

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.19)

where Ec denotes the average of the diabatic excitation energies, theintra-dimer Coulomb coupling strength is denoted by J1 and theinter-dimer Coulomb coupling strength by J2. The diabatic electronicexcitation energies are assumed to be the same within a dimer. Incontrast the excitation energies of neighbouring dimers differ by someconstant energy ΔE. Again four different scenarios are considered,see Tab. 3.2. The scenarios A and B cover the situation that theinter-dimer Coulomb coupling is equal to the intra-dimer Coulombcoupling, i.e. J1 = J2, whereas for C and D the intra-dimer Coulombcoupling is ten times larger than the inter-dimer Coulomb coupling,i.e. J1 = 10J2.The adiabatic electronic level structure is characterized by the

energy of the levels and the amplitude of the exciton states, that isthe square of the expansion coefficient in Eq. (2.22). It features a


0

1

2

3

4

5

0 500 1000 1500 2000

J(ω

) [1

03 cm-1

]ω [cm-1]

b)MBO

Debye

0

50

100

150

200

0 200 400 600

J(ω

) [c

m-1

]

ω [cm-1]

ΔE

ΔE

ΔE

J2

J2

E

Ec

a)

2J1

2J1

2J1

2J1

J2

-200

-150

-100

-50

0

50

100

150

200

1 2 3 4 5 6 7 8

E-E

c [c

m-1

]

diabatic state

J1=-50cm-1c)

-750

-600

-450

-300

-150

0

150

300

450

600

750

1 2 3 4 5 6 7 8

E-E

c [c

m-1

]

diabatic state

J1=-500cm-1d)

Figure 3.7.: (a) Schematic view of the funnel model system. (b)Spectral density models (parameters given in the text and Tab. 3.3).(c) Amplitudes of the exciton states at the different monomer sites forJ1 = J2. (d) Same as (c) for J1 = 10J2

distinct separation into an upper and a lower manifold for C and D,Fig. 3.7 panel d. In contrast to this separation the gap energiesbetween the levels for A and B are of the same order of magnitude,Fig. 3.7 panel c. The adiabatic levels of the upper/lower manifold


Table 3.2.: Intra-dimer Coulomb coupling strengths for various scenar-ios. The inter-dimer Coulomb coupling strength (J2 = −50 cm−1) andthe gap energy (ΔE = 50 cm−1) are the same for all scenarios.

scenario J1 [cm−1] Spectral density modelA -50 MBOB -50 DebyeC -500 MBOD -500 Debye

Table 3.3.: Components of the MBO spectral density for the funnelmodel system.

component ξ Sξ ωξ [cm−1] γξ [cm−1]1 0.197 206.0 100.02 0.215 211.0 100.03 0.037 552.0 100.04 0.208 1371.0 100.05 0.083 1570.0 100.0

for C and D stem from the upper/lower exciton states of the individualdimers. These states are localized on the associated dimers, i.e. theamplitude of the exciton states at the monomers which belong to theother dimers is small. In A and B the adiabatic states are delocalizedover the whole aggregate. Further, two different spectral densities areconsidered for each system configuration, namely a Debye spectraldensity, Eq. (2.110) with η = 0.7 and γ = 100 cm−1 (B and D), anda five-mode MBO spectral density (A and C) which is specified inTab. 3.3. Note that it is again assumed, that all monomers coupleto individual but equal baths and thus the monomeric index m isskipped for the parameters of the spectral densities.


-1000 -500 0 500 1000 1500 2000 2500

norm

aliz

ed in

tens

ity

E-Ec [cm-1]

Figure 3.8.: Absorption spectra for the different scenarios for thefunnel model system (A: red solid line and triangles; B: red dashedline and circles; C: blue solid line and crosses; D: blue dashed line andsquares).

The MBO spectral density is adapted from the spectrum of theintra-molecular vibrations of a perylene bisimide pigment, which isa versatile building block for artificial aggregates used as molecularwires or antennae [76, 81–83]. Whereas the Debye spectral densityhas its maximum around 100 cm−1 and has no significant amplitudefor frequencies higher than 600 cm−1, the MBO spectral density hasa significant amplitude up to 2000 cm−1, Fig. 3.7 panel b.Converged results for B and D were obtained with N = 6 and K = 8,

i.e. one correlation function term per monomer. The results for theMBO scenarios (A and C), which were calculated with N = 4 andK = 80, that is two correlation function terms per mode and monomer,are not fully converged. However, the corresponding hierarchy size


of approximately two million matrices, cf. Eq. (2.78), approaches thelimit of the current version of the Rostock HEOM package, whereas thehierarchy size of approximately 3000 matrices for the Debye scenariosis far below that limit. This indicates the advantages of the Debyespectral density in the context of the HEOM formalism, but notethat this spectral density is not suitable to model structured spectraldensities. The error of the calculations for A and C, which can beestimated via comparison of the shown results with results obtainedwith N = 3, is below ten percent.The absorption spectra, Fig. 3.8, for B and D show only a single

peak at the low energy side of the spectrum, which is specific fornegative Coulomb couplings (J-aggregates). In contrast, the spectrafor A and C feature in addition to the low-energy peak a vibronicprogression due to the vibronic structure implied by the MBO spectraldensity, cf. Sec. 3.1. The stronger red-shift of the spectra for C and Dwith respect to the ones for A and B follows from the larger value ofthe intra-dimer Coulomb coupling J1 in the former cases.In order to study the spatial energy transfer within the aggregate

the initial condition was chosen such that the highest adiabatic excitonstate is populated. This state is mainly located at the first dimer inall scenarios, cf. Fig. 3.7. Note that the spatial energy transfer, that isthe redistribution of the population among the diabatic (local) states,comes along with the relaxation between the adiabatic states.The population dynamics of the dimers, Fig. 3.9, shows in all

scenarios the same general behaviour, i.e. the population slides downin the energy funnel formed by the dimers towards the energeticallylowest dimer. This process is faster for B and D in comparison to Aand C. The difference between the population dynamics for A and B,as well as for C and D, respectively, is not that pronounced. However,the adiabatic populations, Fig. 3.10, show a significantly differentbehaviour for the different scenarios. For A and B the population ofthe states two to six increases to a similar value within a few tens offemtoseconds. This is due to the small energy gap between the states,e.g., approximately 200 cm−1 between state one and six. Both spectraldensities, Fig. 3.7 panel b, feature a significant amplitude in this


0.0

0.2

0.4

0.6

0.8

1.0

0 500 1000 1500

diab

atic

pop

ulat

ion

t [fs]

J1=-50cm-1

MBOρe1e1

+ρe2e2ρe3e3

+ρe4e4ρe5e5

+ρe6e6ρe7e7

+ρe8e8

0 500 1000 1500 2000

t [fs]

J1=-500cm-1

Debyeρe1e1

+ρe2e2ρe3e3

+ρe4e4ρe5e5

+ρe6e6ρe7e7

+ρe8e8

Figure 3.9.: Population dynamics of the dimers, cf. Fig. 3.7, calculatedvia the dynamics of the diabatic one-exciton states for the differentscenarios (A: left panel red lines and squares; B: left panel blue linesand circles; C: right panel red lines and squares; D: right panel bluelines and circles).

frequency region and thus enable an effective relaxation, cf. Sec. 3.1.In contrast, for C and D, where the energy gap between the upper andlower manifold is approximately 800 cm−1, the population is graduallytransferred to the energetically lower states. For C this transfer occursmainly in the lower exciton manifold after a fast relaxation of thepopulation from state one to state five, whereas for D the populationtransfer occurs within the first few hundred femtoseconds in the upperand lower exciton manifolds. The different behaviour in both scenariosis due to the different amplitude of their spectral densities in the high-frequency region, i.e. the MBO spectral density features a significantly


0.0

0.2

0.4

0.6

0.8

1.0

adia

batic

pop

ulat

ion

State 1 State 2 State 3 State 4

0.0

0.2

0.4

0.6

0.8

0 500 1000 1500

adia

batic

pop

ulat

ion

t [fs]

State 5

0 500 1000 1500

t [fs]

State 6

0 500 1000 1500

t [fs]

State 7

0 500 1000 1500 2000

t [fs]

State 8

Figure 3.10.: Population dynamics of the adiabatic one-exciton statesfor the different scenarios (A: red solid lines and triangles; B: red dashedlines and circles; C: blue solid lines and crosses; D: blue dashed lines andsquares). Note that the states are labelled in energetically decreasingorder, cf. Fig. 3.7.

larger amplitude at the transition energy between state one and fivethan the Debye spectral density. Owing to the localization of thesestates on the same dimer, which leads to a strong coherence of thesestates for C, Fig. 3.7 panel d, there exists a pronounced oscillatorybehaviour of the corresponding populations.In order to investigate the influence of the different system and

bath specifications on the exciton delocalization and the populationdynamics, it is instructive to study not only the population of thestates, i.e. the diagonal elements of the reduced density matrix, butalso the off-diagonal elements of the reduced density matrix, the


A: MBO J1=-50cm-1

Re(ρeiej)

i 1 2 3 4 5 6 7 8

j

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

t=0fsB: Debye J1=-50cm-1

i

t=0fsC: MBO J1=-500cm-1

i

t=0fsD: Debye J1=-500cm-1

i

t=0fs

i12345678

j

t=100fs

i

t=100fs

i

t=100fs

i

t=100fs

i12345678

j

t=500fs

i

t=500fs

i

t=500fs

i

t=500fs

1 2 3 4 5 6 7 8i

12345678

j

t=2000fs

1 2 3 4 5 6 7 8i

t=2000fs

1 2 3 4 5 6 7 8i

t=2000fs

1 2 3 4 5 6 7 8i

t=2000fs

Figure 3.11.: Snapshots of the real part of the diabatic density matrixfor various scenarios and propagation times.

so-called coherences. For all scenarios the initial diabatic densitymatrix, Fig. 3.11, is delocalized over several monomers according tothe corresponding eigenvector decomposition, Fig. 3.7 panels c and d.Further, the coherences between the different monomers decay veryfast within the first 100 fs. Whereas for A and B mainly coherencesbetween neighbouring monomers survive, for C and D a strong coher-ence between the monomers which belong to the same dimer develops.

3.3. Fenna-Matthews-Olson complex 75

In contrast to A and C, for B and D additional coherences betweenmonomers of different dimers remain significant. For long propagationtimes these additional coherences vanish and there appears a strictdimerization for C and D, i.e. there are only coherences betweenmonomers which belong to the same dimer. This dimerization is notpresent for A and B, where only coherences between neighbouringmonomers survive.To summarize, the dynamics of the model aggregate depends, sim-

ilarly to the dimer system, cf. Sec. 3.1, crucially on the interplaybetween the system and the bath. In particular, the shape of thespectral density is rather critical for the system dynamics, especiallyif there exist coupled levels with a large energy gap.

3.3. Fenna-Matthews-Olson complexIn the previous sections, the dependence of the system dynamics onthe bath characteristics for a dimer model and a model of a funnel-likeaggregate was investigated using model spectral densities. However,light-harvesting antennae like the Fenna-Matthews-Olson complexfeature a more heterogeneous system structure as well as a complexspectral density. The FMO complex, which serves as an energy funnellinking the light-harvesting chlorosome with the reaction center ingreen sulphur bacteria [1], consists of three identical subunits. Eachof the subunits is formed by an aggregate of seven bacteriochloro-phyll a (BChl a) molecules embedded in a protein environment [84],Fig. 3.12. Note that there exists an eighth BChl a molecule, whichserves as a linker molecule to the chlorosome baseplate [85, 88, 89].This additional molecule changes the weight of the energy pathwaysthrough the FMO complex and might suppress the observed coherentoscillations in the diabatic populations of sites one and two, Fig. 3.12,which are strongly coupled to each other [89]. However, the eighthBChl a molecule will not be considered in the present FMO model.For a detailed study of its influence on the dynamics of the FMOcomplex see Ref. [41].


1

2

34

5

6

7

8

Figure 3.12.: Structure of the monomeric subunit of the FMO com-plex of Prosthecochloris aestuarii consisting of eight BChl a molecules(labelled) embedded in a protein environment of folded β-sheets (blue).The structural data was obtained via X-ray crystallography experi-ments [85] and is available via the RCSB protein data bank (PDB ID:3EOJ) [86]. The figure was created using the VMD program [87].

Due to the structure of the monomeric subunit of the FMO complexand the influence of the protein environment, the Coulomb couplingbetween the BChl a molecules as well as the site energies are, incontrast to the model used in Sec. 3.2, rather heterogeneous. Thisleads to two primary energy transport pathways towards the BChl amolecule with the lowest site energy (site three), that is from site onevia site two to site three and from site five via sites six and seven tosite three. The diabatic one-exciton Hamiltonian, which correspondsto the labeling of the BChl a molecules in Fig. 3.12 and the structural


data from Prosthecochloris aestuarii, is given (in cm−1) by [89]

H(1)s = 1El +

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

310.0 −97.9 5.5 −5.8 6.7 −12.1 −10.3−97.9 230.0 30.1 7.3 2.0 11.5 4.8

5.5 30.1 0.0 −58.8 −1.5 −9.6 4.7

−5.8 7.3 −58.8 180.0 −64.9 −17.4 −64.46.7 2.0 −1.5 −64.9 405.0 89.0 −6.4

−12.1 11.5 −9.6 −17.4 89.0 320.0 31.7

−10.3 4.8 4.7 −64.4 −6.4 31.7 270.0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.20)with El = 12195 cm−1.Although there were attempts to calculate the spectral density

of the FMO complex via molecular dynamics simulations [23], themost reliable available data stems from measurements of the spectraldensity via a combination of fluorescence line narrowing (FLN) andtemperature-dependent linear absorption experiments [9]. Whereasthe fluorescence line narrowing experiments provide access to the shapeof the spectral density, the overall Huang-Rhys factor, i.e. the integralover the spectral density, cf. Eq. (2.114), needs to be evaluated viaabsorption measurements [9]. The FLN experiments were performed ata temperature of 4 Kelvin and the excitation was tuned such that onlythe pigment with the lowest energy has been excited. Therefore, themeasured spectral density represents the monomeric spectral density ofa BChl a molecule in the FMO complex, i.e. the couplings between themolecules are neglected. Further, the measurements do not account forthe heterogeneity of the environment, i.e. it is assumed that all BChl amolecules within the complex have the same spectral density. However,recent calculations [23] show that there are differences between thespectral densities of the individual monomers. Note again that incontrast to Eq. (2.114), the experimentally obtained spectral density


0

0.001

0.002

0.003

0.004

0.005

0 50 100 150 200 250 300

J(ω

) [1

/cm

-1]

ω [cm-1]

a)

12100

12200

12300

12400

12500

12600

12700

1 2 3 4 5 6 7

E [c

m-1

]

diabatic state

b)

Figure 3.13.: (a) Spectral density of a BChl a molecule in the FMOcomplex (black dotted line: experimental data [9], red solid line: fit toMBO spectral density, blue dashed line: fit components). (b) Ampli-tudes of the exciton states at the different monomer sites, cf. Fig. 3.7.

is usually defined such that the overall coupling strength is given by

S =∞∫

0

dωJ(ω), (3.21)

that is both definitions differ by a prefactor of πω2. Thus, to extractthe parameters for quantum dynamics simulations, e.g., via a fit of theexperimental data, one needs to adjust either the experimental data orthe applied fit functions. For example the fit function correspondingto the MBO spectral density, cf. Eq. (2.117), is given by

J(ω) =∑m

∑ξ

Sξ,m

π

ωωξ,mγξ,m

(ω2 − ω2ξ,m)2 + ω2γ2

ξ,m

. (3.22)


Table 3.4.: Components of the extracted MBO spectral density for aBChl a molecule in the FMO complex.

component ξ Sξ ωξ [cm−1] γξ [cm−1]1 0.252 37.6 60.52 0.018 73.5 15.73 0.009 118.5 15.44 0.004 159.6 34.95 0.007 173.2 12.96 0.009 185.9 14.17 0.009 195.3 12.18 0.006 238.3 12.09 0.002 261.8 10.410 0.005 285.2 11.7

The measured spectral density of the BChl a molecules in the FMOcomplex, Fig. 3.13 panel a, was parametrized via a fit to a ten modeMBO spectral density. Note that the measured spectral density isnormalized such that its integral corresponds to the Huang-Rhysfactor of 0.45 [9]. Further, it is assumed that the spectral densityis the same for all BChl a molecules in the complex and thus themonomer index m can be skipped. The obtained fit parameters arelisted in Tab. 3.4. Note that the fit of the measured spectral densityis quite accurate except for the frequency range below 50 cm−1 wherethe double-peak structure of the experimental data is approximatedby a single MBO component. However, this approximation should besufficient as the energy gaps between the adiabatic states, Fig. 3.13panel b, which can be associated with the first main energy pathway,i.e. state six, three and one, are larger than 100 cm−1.To investigate the dynamics of the system corresponding to the first

main energy pathway, namely from site one via site two to site three,the initial condition was chosen such that the first diabatic state is


0.0

0.2

0.4

0.6

0.8

1.0

diab

atic

pop

ulat

ion

1234567

0.0

0.2

0.4

0.6

0.8

0 200 400 600 800 1000

adia

batic

pop

ulat

ion

t [fs]

Figure 3.14.: Diabatic and adiabatic population dynamics of the FMOcomplex after an initial population of the first diabatic state.

populated. This corresponds to an initial population of the sixthand third adiabatic state, cf. Fig. 3.13 panel b. Note that the resultsare not converged for the given hierarchy setup (N = 3, K = 140)but a larger hierarchy cannot be treated with the current versionof the Rostock HEOM package. Again, the error was estimated viacomparison with a calculation with a lower convergence level. It isapproximately twenty percent.The dynamics of the diabatic population, Fig. 3.14, shows a pro-

nounced oscillation between populations of the first and second dia-batic state, whereas the population of the third diabatic state, as wellas those of the states four to seven, shows an almost linear behaviour.This is in agreement to the population dynamics reported in the litera-ture [89, 90]. The frequency of the oscillation, which is approximately250 cm−1, does not fit to the energy gap between the correspondingadiabatic states six and three, cf. Fig. 3.13 panel b, which is 210 cm−1.


This indicates, in combination with the oscillations of the adiabaticpopulation of states six and three, Fig. 3.14, that the oscillation isof vibronic origin, cf. Sec. 3.1. The small population of the diabaticstates four to seven and the corresponding adiabatic states two, four,five and seven point towards the possibility to model the dynamics ofthe FMO complex by a reduced three-site model, which would reducethe numerical effort by far. A comparison of the full FMO model withsuch a reduced model, including the first, second and third site, isgiven in Ref. [41]. Note that the reduced model ansatz is valuableto study individual transfer pathways, but might by insufficient tostudy the dynamics after an optical excitation. The latter would leadto an initial excitation of various monomers and thus also pathways,which are not covered by a reduced model, might be important. As itwas mentioned in Sec. 3.1, the connection of the oscillations in thepopulation dynamics to the oscillations observed in 2D-spectroscopyexperiments [16] is non-trivial and needs to be revealed by furtherinvestigations.

4. Summary

Since the observation of long lasting oscillations in the 2D-spectraof the FMO complex in 2007 [16], the influence of coherence on theexciton dynamics in natural photosynthetic antennae, which waslong considered to be of hopping-like nature sufficiently describedby rate equations, is of peculiar interest in the context of artificialphotosynthesis. In order to study the dynamics, sophisticated meth-ods, which are able to describe the dissipative quantum dynamicsof molecular aggregates embedded in a protein environment ratheraccurately, are required. The HEOM formalism, outlined in Sec. 2.2.2and implemented, e.g., in the Rostock HEOM package, provides sucha powerful tool. Here, the influence of the environment is mimickedvia an in principle infinite hierarchy of auxiliary density matrices.This leads to a considerable numerical effort even for small systems,which restricts the applicability of the HEOM method. However,recent improvements, concerning the method itself as well as thecomputational algorithms, facilitate the treatment of the dynamics oflight-harvesting complexes.In the present master thesis, the exciton dynamics of a dimer system,

a model aggregate resembling light-harvesting antennae and the FMOcomplex were investigated using the HEOM method. The populationdynamics, as well as the linear and 2D-spectra, of the dimer shows astrong dependence on the system and bath properties. In particular,the value of the spectral density at the transition energies of the systemis crucial for the relaxation dynamics, as it reflects the coupling ofthe corresponding transition to bath. The oscillatory features, whichappear in the population dynamics, can be interpreted using theadiabatic vibronic level structure of the dimer. Unfortunately, thisstructure is not available for larger aggregates. However, the origin


84 4. Summary

of the oscillatory features can be determined by a comparison of theadiabatic and diabatic electronic populations. Oscillations, which stemfrom a superposition of vibronic levels, are present in both, whereasones of electronic origin are only present in the latter. The connectionof the oscillations in the populations to the ones in 2D-spectra needsto be investigated in more detail in the future.Similar to the dimer system, the dynamics of the model aggregate

is determined by the interplay of the system and the bath. Especiallythe weight of the transfer pathways within the aggregate dependscrucially on the shape of the spectral density. The dynamics of theFMO complex, which was calculated using experimental data for thespectral density, indicates that coherences might play an importantrole in some of the transfer steps. In particular, the transfer stepfrom the first to the second site seems to be governed by a coherence,which can be assigned by a comparison of the diabatic and adiabaticpopulations to be of vibronic origin.

A. The Rostock HEOM packageThis appendix describes the Rostock HEOM package which wasdeveloped during the present master thesis. The package is availableupon request. First, an overview of the package itself and somefeatures of the implementation will be given. Second, the installationand the usage of the main program and some analysis tools will bedescribed. Finally, the appendix contains the input documentation ofthe package.

A.1. OverviewThe Rostock HEOM package is written in FORTRAN 90/95 and thesubroutines are organized in several modules, Tab. A.1, correspondingto their function. In general the program execution consists of fourparts, Fig. A.1. First, the input file is read by the program. Second,the system and bath parameters are processed, i.e. all operators aretransformed to the eigenbasis of the Hamiltonian, the correlationfunction is calculated for the given spectral density, and the hierarchyis constructed. Third, the propagation corresponding to the definedtask, i.e. the calculation of the population dynamics, the linear or2D-spectra, is performed in the excitonic eigenbasis. Finally, thecalculated values and a summary are printed to the output files. Notethat some values are continuously printed, i.e. the population of thestates or the autocorrelation function. The modules are designed to beindependent of each other, i.e. the data is in general passed via globalvariables. Exceptions from this concept are made for the propagationroutines.The bath correlation function needs to be parametrized to fulfill the

requirements of the HEOM formalism, cf. Sec. 2.2.2. So far the pack-

M. Schröter, Dissipative Exciton Dynamics in Light-Harvesting Complexes,BestMasters, DOI 10.1007/978-3-658-09282-5,© Springer Fachmedien Wiesbaden 2015

86 A. The Rostock HEOM package

read input file

process systemdata

process bathdata

build hierarchy

populationdynamics

linear spectra 2D-spectra

output results

6

7 8

9

10,11,14,15 4,10,11,12,14,16 4,10,11,12,14,17

13,15,16,17

prep

arat

ion

prop

agat

ion

Figure A.1.: General structure of the current implementation. Theitalic numbers refer to the main modules, which are involved in thecurrent step, cf. Tab. A.1.

age is able to process the MBO, Debye and Ohmic spectral density,cf. Eq. (2.117), Eq. (2.110) and Eq. (2.109). Whereas the correspond-ing correlation functions for the MBO and the Debye model are givenby Eq. (2.119) and Eq. (2.111), respectively, the one correspondingto the Ohmic spectral density needs to be evaluated, e.g., via theMeier-Tannor parametrization scheme, cf. Eq. (2.126). Note that the

A.1. Overview 87

Table A.1.: List of modules (as of 2014-06-30).

module No. descriptiontypes_and_constants.f90 1 Data type definition, conversion

factors and physical constants.profiling.f90 2 Routines for a basic run time

profiling of the code.speed.f90 3 Routines to log the time needed

for the individual propagationsteps.

fft.f90 4 Interface to the FFTW3 libraryroutines.

global_variables.f90 5 Declaration of global variables.read_input_sections.f90 6 Routines to process the input

file.eigensystem.f90 7 Routines to calculate the eigen-

system of the Hamiltonian viaLAPACK routines and to trans-form all matrices and operatorsto the eigenbasis of the Hamil-tonian.

poles of the Bose-Einstein distribution function, cf. Eq. (2.106), aretreated via the Matsubara scheme. All correlation functions obey therequired general form

C(t) =K∑

k=1ck e−γkt + δC. (A.1)

An explicit treatment of many expansion terms results in a hugehierarchy which is not desirable at all. Therefore, only a small numberof expansion terms, which needs to be specified in the input, will betreated explicitly. However, the residual part of the correlation func-


Table A.2.: Continued list of modules (as of 2014-06-30).

module No. descriptioncorrelation_function.f90 8 Routines to calculate the cor-

relation function expansioninto a sum of exponentials forthe MBO Debye and Ohmicspectral density.

build_hierarchy.f90 9 Routines to build the requiredhierarchy, i.e. label the ADM,build the references tablesand initialize the matrices.

rhs_heom.f90 10 Routine which evaluates theright-hand side of the HEOM.

ontheflyfiltering.f90 11 Routines for a numerical fil-tering of the hierarchy to re-duce the computational ef-fort.

spectra.f90 12 Routines to calculate the lin-ear and nonlinear spectra.

summary.f90 13 Routines to create the log file.propagation_algorithms.f90 14 Propagation routines.population_dynamic.f90 15 Execution routines to calcu-

late the population dynamics.linear_signal.f90 16 Execution routines to calcu-

late the linear spectrum.nonlinear_signal.f90 17 Execution routines to calcu-

late the nonlinear spectrum.

tion can be approximated following the method outlined in Ref. [36],i.e.

δC =K∑

k=K+1ck e−γkt + δC ′, (A.2)

A.1. Overview 89

employing all terms up to a specific number K. This approximationyields for the coefficient Δ, cf. Eq. (2.82),

Δ =K∑

k=K+1

ck

γk. (A.3)

Note that the coefficient Δ is determined for each correlation functionCζ(t), if there exist multiple environments connected to the samesystem.The process of labeling the ADM (RDM) is divided into two parts.

First, the algorithm follows the K-dimensional quasi-tree structurecorresponding to the hierarchy in pre-order setup (root-left-rightlabeling). The quasi-tree will be set up recursively and direct knot -leaf references will be created immediately. However, following thealgorithm there will be some references, which are not created directly.These references will be added in a second step, which is the actualbottleneck of this algorithm. The labeling scheme is sketched inFig. A.2.The propagation algorithms, i.e. the Runge-Kutta 4 and the Runge-

Kutta-Fehlberg 4/5 algorithm, require multiple copies of the ADMarray to perform one propagation step. Therefore, beside the largeramount of time needed to propagate a larger hierarchy, the availablememory limits the hierarchy size that can be treated with the currentversion of the package. The propagation routines are parallelizedusing two different schemes. There exists a shared parallelization viaOpenMP, which is suited, e.g., for machines with rather low memory,and a parallel memory parallelization via MPI. The parallelization typecan be chosen in the "make" file during the installation of the package,cf. App. A.2. As the calculation of 2D-spectra via the responsefunction formalism requires a large number of propagations thereexists also a parallel memory parallelization for the correspondingroutine.The hierarchy is automatically numerically filtered by a on-the-

fly truncation algorithm, if the keyword FILTERING is given inthe input. In the present implementation, all ADM which have no


ρ00

ρ10 ρ01

ρ20 ρ11 ρ02

1

2 5

3 4 6

Figure A.2.: Scheme of ADM labeling system for N = 2 and K = 2.The labels of the ADM (red numbers) as well as the direct references(black lines) are created via running trough the quasi-tree correspondingto the hierarchy in pre-order setup. The indirect references which donot appear in the quasi-tree (red lines) are created by an additionalroutine.

element which is bigger than 10−8 are deleted from the hierarchy andthe corresponding references are eliminated. The truncation routineis executed after every hundredth integration step.

A.2. Installation and usage of the programsPlease note that it is necessary to edit the compiler and library setupin the "make" file to adjust the program to the local environment.Note that some of the required libraries, that are FFTW3, LAPACK,

A.2. Installation and usage of the programs 91

and BLAS, might be included in more sophisticated libraries like theIntel Math Kernel Library (MKL).The program makes use of both parallel and shared memory paral-

lelization. Shared memory parallelization via OpenMP is used for theintegrator routines and will be enabled by a compiler flag (e.g. -openmpfor ifort). Parallel memory parallelization might either be used forparallelization of the integrator routines (compiler flag -Dmpi_int) orthe 2D-spectra calculation (-Dmpi_2D). Note that mpi parallelizationwill be in general controlled by a pre-compiler (e.g. -fpp for the ifortcompiler). To compile the programs it is recommended to execute theinstallation script install.sh. This script will create the executables.It is possible to add the executables to your shell environment viapassing the location and name of your shell configuration file to thescript, e.g.

./install.sh /.bashrc

The main HEOM program needs to be called with an input file asargument.

HEOM inputfileor

mpirun -np N_proc HEOM inputfile

For a detailed description of the requirements of the input file seeApp. A.3.Two analysis programs are provided to calculate linear and non-

linear spectra, respectively. They need to be called with severalarguments which are listed below.

linspecHEOM inputfile outputfile numberofsteps stepwidthleftoutputborder rightoutputborder unitsystem abs/emi

2DspecHEOM inputfile outputfile xsteps ysteps xstepwidth ystepwidthxdirection ydirection leftoutputborder rightoutputborder unitsystem


As unit systems so far eV, cm−1 and nm are implemented. abs/emirefers to absorption and emission for the linear spectrum (note thatthe latter is not implemented so far), x − /y − direction indicates thedirections of the underlying Fourier transform. The step width needsto be given in fs. Note that there exist two utility scripts 2Dspec.shand joindat.sh which facilitate the calculation of the 2D-spectra.

A.3. Input documentationA.3.1. Run sectionThe run section contains all general information needed for the prop-agation, like the dimensionality of the reduced density matrix andall operators, the integration scheme and so on, as well as keywordsto enable features like the on-the-fly filtering of the hierarchy. It isspecified by

run-sectionkeywords (each in a single line) #commentsend-hierarchy-section

The keywords are listed below.

Table A.3.: List of run-section keywords (as of 2014-06-30)

keyword descriptionNSTATE = N required; specifies the dimensionality of the

reduced density matrices and the operators,the value N needs to be a positive integernumber

NSTEP = N required; specifies the number of output steps(which may differ from the number of inte-grator steps), the value N needs to be apositive integer number; NOT required for2D-spectra calculations

A.3. Input documentation 93

Table A.4.: Continued list of run-section keywords (as of 2014-06-30)

keyword descriptionTOUT = R required; specifies the output time

step in fs, the value R needs tobe a positive real valued numberand an integer multiple of tint;NOT required for 2D-spectra cal-culations

DEBUG optional; if this keyword is given,the list of references and the RDMlabels will be written to separatelog files

THREADS = N optional; specifies the number ofthreads used for parallelization ofthe integration scheme, the valueN needs to be a positive integernumber; note that to avoid over-load the number of threads will beadjusted by the program, never-theless the value of N is the maxi-mum number of possible threadsused by the program

TEMPERATURE = R required; specifies the temperaturein K of the system, the value Rneeds to be a positive real valuednumber

TRUNCORDER = N required; specifies the depth of thehierarchy, the value N needs to bea positive integer number

DELTAAPPROX optional; enables the approxima-tion of the residual part of thebath correlation function by a δ-function, cf. Eq. (2.82)



keyword descriptionFILTERING optional, not available for 2D-

spectra calculations; enablesthe on the fly numerical filter-ing algorithm which deletesall reduced density matriceswhich contains only elementsthat are smaller than 10−8

from the hierarchy (compare[35, 36]); the adjustment rou-tine is called after each 100integrator steps

ADIABATICINITIALRDM optional; declares that thesystems reduced density ma-trix given in the initial RDMsection is already representedin the adiabatic basis

INTEGRATOR = C;R1;R2 required,required,optional;specifies the integrator usedto solve the HEOMs; thefirst argument denotes theintegration scheme, thesecond one the initial inte-gration step size and the lastone the error tolerance foradaptive schemes, Runge-Kutta 4 (C = RK4) and theRunge-Kutta-Fehlberg 4/5(C = RKF4) schemes areimplemented



keyword descriptionSPECTRARANGE = R1;R2;C optional; specifies the range

and unit system for spec-tra output; R1 denotes theleft plot border, R2 theright plot border and C(cm-1, nm, eV ) the unit sys-tem, default are 0;50000;cm-1

ADIABATICCOUPLING optional; declares that thesystem-bath interaction oper-ators given in the interactionsection are represented in theadiabatic basis, which meansthat the bath couples to theadiabatic states directly

A.3.2. Task sectionThe tasks section contains the information which tasks shall be per-formed by the program. It is specified as follows.

task-sectionkeywords (each in a single line)end-tasks-section

The keywords are listed below.

Table A.7.: List of task-section keywords (as of 2014-06-30)

keyword descriptionpopulation outputs the population of the diabatic and

adiabatic states to the file pop.dat


Table A.8.: Continued list of task-section keywords (as of 2014-06-30)

keyword descriptionautocorrelation outputs the autocorrelation function to the

file auto.dat; note that a dipole operator isrequired to perform this task

absorptionspectrum sets autocorrelation and calculates thelinear absorption spectrum with thelinspecHEOM algorithm; output will bewritten to spec.dat; note that a dipole op-erator and a specified plot-section are re-quired

emissionspectrum sets autocorrelation and calculatesthe linear emission spectrum with thelinspecHEOM algorithm; output will bewritten to spec.dat; note that a dipoleoperator and a specified plot-section arerequired

2Dspectrum calculates third order response functions(see Ishizaki and Tanimura, J. Chem. Phys.125, 084501 (2006)) which are required tocalculate the 2D-spectrum, a dipole opera-tor as well as the 2D section are required.

2Dspectrumdetail calculates the individual components of thethird order response functions (see Chenet al. J. Chem. Phys. 132, 024505 (2010),eq. 15) which are required to calculate the2D-spectrum, a dipole operator as well asthe 2D section are required.

A.3.3. Parameter sectionThe parameter section contains all parameter definitions which mightbe used in the Hamiltonian, dipole, interaction, bath and initial-RDM


section. Each parameter should be written in a single line. The sectionis specified as follows.

parameter-sectionidentifier=value:unit #commentsend-parameter-section

The identifiers must not be longer than 16 characters. So far as unitselectron volts (eV) and wave numbers (cm−1) are supported. Notethat all values without a unit (skip the colon) will be handled as theywere in atomic units.

A.3.4. Hamiltonian sectionThe Hamiltonian section defines the system Hamiltonian in diabaticrepresentation. Each element of the Hamiltonian should be written ina single line. The section is specified as follows.

hamiltonian-sectionidentifier [space] i [space] j [space] value/parameterend-hamiltonian-section

There are two possible identifiers. S denotes symmetric matrix ele-ments, that means Mij = Mji = value, and A denotes just thematrix elements Mij . The value can be specified either directly orvia a parameter identifier specified in the parameter section. All lineswhich do not start with one of the identifiers will be ignored and canbe seen as comments. Note that all matrix elements which are notspecified will be zero.

A.3.5. Dipole sectionThe dipole section defines the dipole operator in diabatic represen-tation. Each element of the dipole operator should be written in asingle line. The section is specified as follows.


dipole-sectionidentifier [space] i [space] j [space] value/parameterend-dipole-section

There are two possible identifiers. S denotes again symmetric matrixelements of the type Mij = Mji = value and A denotes just thematrix elements Mij . The value can be specified either directly orvia a parameter identifier specified in the parameter section. All lineswhich do not start with one of the identifiers will be ignored and canbe seen as comments. Note that all matrix elements which are notspecified will be explicitly zero.

A.3.6. Interaction sectionThe interaction section defines the system-bath interaction operatorsQ. Each element of a specific operator should be written in a singleline. The section is specified as follows.

interaction-sectionusedens=Nidentifier [space] i [space] j [space] value/parameterend-interaction-section

You can specify an arbitrary number of operators. Note that eachoperator will be connected to a spectral density given in the bathsection by the keyword usedens = N . N refers to the number of adefined spectral densities. It is possible to connect multiple operatorsto one spectral density, e.g., couple multiple monomers to the samekind of bath. There are two possible identifiers for the matrix elements.S denotes symmetric matrix elements Mij = Mji = value and Adenotes just the matrix elements Mij . The value can be specifiedeither directly or via a parameter identifier specified in the parametersection. All lines which do not start with one of the identifiers will beignored and can be seen as comments. Note that all matrix elementswhich are not specified will be explicitly zero.


A.3.7. Bath sectionThe bath section contains the definitions of the spectral densities.Each spectral density component should be written in a single line.The section is specified as follows.

bath-sectionidentifier [space] parameter list [space] Nend-bath-section

The identifier might either be debye, ohmic or brownian. The specificparameter lists are explained below, cf. Sec. 2.3 for the definition ofthe individual parameters. Note that you can use parameters definedin the parameter section. N defines the number of frequencies whichwill be explicitly handled within the HEOM formalism.

Table A.9.: List of supported spectral densities with parameters (asof 2014-06-30)

identifier parameter listdebye ηm [space] ωc,m [space] N

ohmic ηm [space] ωc,m [space] N

brownian 2Sξ,m [space] ω(ξ)c,m [space] ωξ,m [space] N

A.3.8. Initial-RDM sectionThe initial-RDM section defines the initial top level reduced densitymatrix in diabatic representation. Each component of the initial RDMshould be written in a single line. The section is specified as follows.

initial-RDM-sectionidentifier [space] i [space] j [space] value/parameterend-initial-RDM-section


The initial top level RDM is given explicitly (all other RDMs will be setto zero). There are two possible identifiers for the matrix specification.S denotes symmetric matrix elements Mij = Mji = value and Adenotes just the matrix elements Mij . The value can be specifiedeither directly or via a parameter identifier specified in the parametersection. All lines which do not start with one of the identifiers will beignored and can be seen as comments. Note that all matrix elementswhich are not specified will be explicitly zero.

A.3.9. 2D sectionThe 2D section defines the propagation times for the 3rd order responsefunctions. It is specified as follows.

2D-sectiont1=t1_start;t1_stepwidth;number_of_t1_stepst2=t2_start;t2_stepwidth;number_of_t2_stepst3=t3_start;t3_stepwidth;number_of_t3_stepsend-2D-section

The _start and _stepwidth values need to be defined in fs. Thenumber of steps is an integer value.

B. Example inputs

This appendix provides example inputs for the Rostock HEOM pack-age, cf. App. A, for the calculations presented in Sec. 3.1. Note thatthe inputs are written for the current version of the Rostock HEOMpackage (as of 2014-06-30), upwards compatibility with future versionsis not granted. The numbers in front of the input keywords denotesthe line numbers of the input file.


102 B. Example inputs

B.1. Population dynamics dimer system1 run-section2 TEMPERATURE=300.03 TRUNCORDER=94 NSTATE=25 TOUT=1.06 NSTEP=40007 INTEGRATOR=RKF4;0.05;1.0e-58 THREADS=89 FILTERING10 DELTAAPPROX11 ADIABATICINITIALRDM12 end-run-section1314 task-section15 population16 end-task-section1718 parameter-section19 E1=10000.0:cm-120 E2=10500.0:cm-121 J12=250.0:cm-122 eta=0.1 # Note eta=2S23 gamma=200.0:cm-124 omega=500.0:cm-125 end-parameter-section2627 hamiltonian-section28 S 1 1 E129 S 2 2 E230 S 1 2 J1231 end-hamiltonian-section3233 bath-section

B.1. Population dynamics dimer system 103

34 brownian eta gamma omega 235 end-bath-section3637 interaction-section38 usedens=139 S 1 1 1.040 usedens=141 S 2 2 1.042 end-interaction-section4344 initialRDM-section45 S 2 2 1.046 end-initialRDM-section


B.2. Linear absorption spectrum dimer system1 run-section2 TEMPERATURE=300.03 TRUNCORDER=94 NSTATE=35 TOUT=0.256 NSTEP=660007 INTEGRATOR=RK4;0.058 THREADS=89 FILTERING10 DELTAAPPROX11 SPECTRARANGE=5000.0;15000.0;cm-112 end-run-section1314 task-section15 absorptionspectrum16 end-task-section1718 parameter-section19 E0=0.0:cm-120 E1=10000.0:cm-121 E2=10500.0:cm-122 J12=250.0:cm-123 eta=0.124 gamma=200.0:cm-125 omega=500.0:cm-126 end-parameter-section2728 hamiltonian-section29 S 2 2 E130 S 3 3 E231 S 2 3 J1232 end-hamiltonian-section33

B.2. Linear absorption spectrum dimer system 105

34 bath-section35 brownian eta gamma omega 236 end-bath-section3738 interaction-section39 usedens=140 S 2 2 1.041 usedens=142 S 3 3 1.043 end-interaction-section4445 dipole-section46 S 1 2 1.047 S 1 3 1.048 end-dipole-section4950 initialRDM-section51 S 1 1 1.052 end-initialRDM-section


B.3. 2D-spectrum dimer system1 run-section2 TEMPERATURE=300.03 TRUNCORDER=54 NSTATE=35 INTEGRATOR=RK4;0.056 THREADS=17 DELTAAPPROX8 FILTERING9 end-run-section1011 parameter-section12 E0=0.0:cm-113 E1=10000.0:cm-114 E2=10500.0:cm-115 J12=250.0:cm-116 eta=0.117 gamma=200.0:cm-118 omega=500.0:cm-119 end-parameter-section2021 task-section22 2Dspectrum23 end-task-section2425 bath-section26 brownian eta gamma omega 227 end-bath-section2829 interaction-section30 usedens=131 S 2 2 1.032 usedens=133 S 3 3 1.0

B.3. 2D-spectrum dimer system 107

34 end-interaction-section3536 hamiltonian-section37 S 1 1 E038 S 2 2 E139 S 3 3 E240 S 2 3 J1241 end-hamiltonian-section4243 dipole-section44 S 1 2 1.045 S 1 3 1.046 end-dipole-section4748 initialRDM-section49 S 1 1 1.050 end-initialRDM-section5152 2D-section53 t1=0.0;0.5;102454 t2=0.0;25.0;4155 t3=0.0;0.5;102456 end-2D-section

C. Electronic vs. vibronic basis

This appendix focusses on the connection of the electronic and vibronicbasis in the adiabatic and diabatic representation for a dimer system.The one-exciton Hamiltonian of a dimer system in the electronicdiabatic representation is given by, cf. Eq. (2.6),

H(1,el)dia =

(Ee1 J12

J21 Ee2

), (C.1)

with J12 = J21 = J . Diagonalization of this Hamiltonian yields theadiabatic electronic Hamiltonian

H(1,el)adia =

(E1 00 E2

), (C.2)

where the adiabatic energies E1 and E2 are given by Eq. (2.23). Therelation between the adiabatic and diabatic electronic Hamiltoniancan mathematically expressed as

H(1,el)adia = S−1

el H(1,el)dia Sel. (C.3)

Here the transformation matrix Sel is given by the eigenvectors ofthe diabatic Hamiltonian. Their elements can be identified as theexpansion coefficients c

(a)α,m in Eq. (2.22).


110 C. Electronic vs. vibronic basis

The diabatic vibronic Hamiltonian including the zeroth and firstvibrational level at each site, cf. Sec. 3.1, is given by

H(1,vib)dia =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Ee100 JF 0000 0 0 JF 10

00 JF 0100 0 JF 11

00

JF 0000 Ee200 JF 10

00 JF 0100 0 0 JF 11

00 0

0 JF 0010 Ee110 0 JF 10

10 JF 0110 0 JF 11

10

0 JF 0001 0 Ee101 JF 10

01 JF 0101 0 JF 11

01

JF 0010 0 JF 10

10 JF 0110 Ee210 0 JF 11

10 0

JF 0001 0 JF 10

01 JF 0101 0 Ee201 JF 11

01 0

0 JF 0011 0 0 JF 10

11 JF 0111 Ee111 JF 11

11

JF1100 0 JF 1011 JF 01

11 0 0 JF 1111 Ee211

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (C.4)

where Eaμν denotes the energy corresponding to the diabatic aggregatestate a with the vibrational excitation μ/ν in the first/second monomerand J the Coulomb coupling matrix element. F

μ′a′ ν′

b′μaνb represents

the vibrational overlap (the Franck-Condon factors) between thevibrational configurations of the different aggregate states [79, 80].The overall coupling elements are given by

〈manbμaνb| H(1,vib)dia

∣∣ma′nb′μ′a′ν ′

b′⟩=J(manb, ma′nb′)

⟨μa|μ′

a′⟩ ⟨

νb|ν ′b′⟩

=JFμ′

a′ ν′b′

μaνb . (C.5)

Note that the Franck-Condon overlap integral, which is given for asingle vibrational mode by

⟨μg|μ′

e

⟩= e−S

μg∑M=0

μ′e∑

N=0

(−1)N √2SM+N

M !N !

×√

μg!μ′e!

(μg − M)!(μ′e − N)!δμg−M,μ′

e−N , (C.6)

depends on the Huang-Rhys factor S. Similar to the diabatic electronicHamiltonian, the diabatic vibronic Hamiltonian can be diagonalized

111

introducing the transformation matrix Svib yielding the adiabaticvibronic Hamiltonian, i.e.

H(1,vib)adia = S−1

vibH(1,vib)dia Svib. (C.7)

However, for small Huang-Rhys factors (S ≤ 0.05), i.e. the Franck-Condon overlaps of the the type F

μgνeμeνg /F

μeνgμgνe are close to unity and

all other close to zero, the upper left part of the adiabatic vibronicHamiltonian can be approximated by the corresponding adiabaticelectronic one. For larger Huang-Rhys factors (S ≥ 0.05) the mixingwith the vibrationally excited vibronic states becomes important.

C. Electronic vs. vibronic basis

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